This course enables students to develop an understanding of mathematical concepts related to algebra, analytic geometry and measurement and geometry through investigation, the effective use of technology, and abstract reasoning. Students will investigate relationships, which they will then generalize as equations of lines, and will determine the connections between different representations of a linear relation. They will also explore relationships that emerge from the measurement of thee dimensional figures and two dimensional shapes. Students will reason mathematically and communicate their thinking as they solve multi step problems.
Part of a baking recipe calls for 600 mL of flour and 5 tablespoons of chocolate chips to make 30 chocolate chip cookies. How much flour and how many tablespoons of chocolate chips would be needed to make 58 cookies? Round your answers to the nearest whole number.
Solution
Hint Clear Info
Flour = mL
Chocolate = tablespoons
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Find the amount of flour and chocolate for 1 cookie:
And ...
Use this to calculate the flour and chocolate for 58 cookies by multiplying each:
Determine the amount of flour:
Nigel's favourite coffee company says to use 10.0 grams of coffee for every 170 mL of hot water. How much coffee would it take, in grams, to make a big 850 mL jug of coffee?
Organic bananas cost $0.10 per 100 g. If a large bunch of bananas costs $4.00 how much does it weigh?
A cool car can drive 100 km on just 4.0 L of gasoline. If 1.0 L of gasoline costs $1.25 then how much would it cost to drive a distance of 500 km from Montreal to Toronto?
Proportions: Dimensions and Area
A print of an ancient map needs to be enlarged according to the table below. Determine the new height, x, and the new area, y.
Solution
Width (cm)
Height (cm)
Area (cm^{2})
Original Size
20
30
600
Enlarged Size
100
x
y
Hint Clear Info
x = y =
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cm
Hint Unavailable
First determine the expansion factor (= 100/20 = 5).
Then, determine the new height (= 30 × 5 = 150cm).
Then, determine the new area (= 100 × 150 = 15,000).
x = 150cm and y = 15,000cm
Formulas with Single Degree Variables
Solve the following.
If the area of a soccer field is 8,000m^{2} and the width is 100m, find the length of the field.
Solution
Hint Clear Info
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m
Hint Unavailable
The circumference of a circle is 29.83cm. What is the radius?
Solution
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cm
Hint Unavailable
Determine the length of a rectangle with a perimeter of 37m and a width of 8.5m.
Solutionp = 2(length) + 2(width)
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m
Hint Unavailable
If the area of a ginormous triangle is 50m^{2} and the height is 5m, determine the dimension of its base.
Solution
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m
Hint Unavailable
If a store sells its slow-cooked meat sandwiches for $9.50 each and made $2,156.50 on sandwiches during the whole day, how many sandwiches were sold?
Solution
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sandwiches
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Solve Problems Modelled with First-Degree Equations
Solve the following questions using your knowledge of variables and rearranging equations.
David has qualified for a Triathlon. He is a strong cyclist and runner, but he is worried about the swim which is 3,900m in length. His goal is to complete the swim with an average speed of 39m/min. If he starts the swim at 1:00pm, how far will he have left in the swim by 2:00pm?
Solution
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m
Hint Unavailable
David does this in 1 hour. Convert to minutes = 60 minutes
Subtract this distance by the total to get the amount left:
A fancy car salesman makes $16/hour plus a $900 bonus for each car they sell. If the person made $2568 over a 48 hour work week, how many cars did they sell?
Solution
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cars
Hint Unavailable
The money they made from the hourly rate is $768
Subtracting $768 from the total, $2568 gives the amount of money made from bonuses, $1800
Dividing the total money from bonuses, $1800 by the amount per bonus, $900, gives the amount of cars sold, 2.
Solving Expressions
Solve the expression below given, y = 2 and x = -2.
Solution
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Substitute the 'x' and 'y' values into the expression,
Simplifying Expressions and Solving Equations
Simplify the expressions, or solve the equations where applicable.
Define the following types of numbers: rational, irrational, natural, whole, and integer.
Solution
Word Problems
Number Problems
The following equation correctly models the sentence, ten times a number is 25 less than 5 times a number. Solution
True.
Write an expression for three times more than a number, using 'x' as your variable.
Solution
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3x
The correct expression for two consecutive odd integers or two consecutive even integers is always
An equation for twice a number is equal to four less than three times a number.Solve for x. Show your work.
Solution
x =
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Number Problems
Write a let statement defining the variable, make an equation, and solve for the unknown lower number. Show your work.
The sum of two numbers is seventy. One of the numbers is 10 more than 2 times the other number.
Solution Video
x =
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Let 'x' represent the first number. Let (2x + 10) represent the other number.
The sum of two consecutive even integers is 58. Determine the larger of the two numbers.
Solution
x =
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Let 'x' represent the first number. Let (x + 2) represent the other number.
The sum of three consecutive odd integers is 489.
Solution
x =
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Let 'x' represent the first, (x + 2) represent the second, and (x + 4) represent the third number.
A larger number is 7 more than a smaller number. Three times the larger number plus 4 times the smaller number equals 42.
Solution
x =
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Let 'x' be the smaller number and (x + 7) be the larger number.
Age Problems
If the sum of two ages is 10, and you let 'x' represent one of the ages, which of the following expressions would represent the other age?
Solution
The other age is the sum (10) minus one of the ages, 'x'
= 10 - x
David is 'x' years old. Sandra is (x + 5) years old. Sandra is 4 times older than David. The correct equation for this statement is:
Solution
It should be:
David's age (x) is lower, so you need to multiply (x) by 4 to equal Sandra's higher age.
Determine the correct equation to solve the following statement. Sarah's aunt is 50 years older than her. How old is Sarah if her aunt is also 3 times Sarah's age? Solution
Sarah is 25, and her aunt is 75 years old.
Age Problems
Write a let statement defining the variable, make an equation, and solve for the unknown. Show your work.
Simon's aunt is 6 times his age. How old is Simon if his aunt is 40 years older than him?
Solution
Simon:
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years old
Hint Unavailable
Let 'x' represent Simons age, and (x + 40) his aunt's age.
Simon is 8 and his aunt is 48 years old.
Michael is younger than Amanda. Four times Amanda's age plus Michael's age is 84. How old is Amanda if the sum of their ages is 30?
Solution
Amanda:
Hint Clear Info
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Let 'x' represent Amanda's age, and (30 - x) Michael's age.
Amanda is 18 years old. (Michael is 12).
If Oliver is x years old and Laila is 2 times older than Oliver, determine an expression for both of their ages in 5 years from now.
Age Now
Age in 5 years
x
2x
Age Problems
Simon is 20 years older than Jackie. 5 years from now Simon will be twice as old as Jackie. How old is Jackie now?
Correct the mistake in the equation for this statement.
Solution
Let 'x' represent Jackie's age. Let (x + 20) represent Simon's age.
For "5 years from now", add 5 to 'x' and add 5 to (x + 20).
Let 'x' represent Jackie's age. Let (x + 20) represent Simon's age
Money Problems
Determine the correct equation to solve the statement below.
The number of ten dollar bills a bank teller has is 8 less than 3 times the number of twenty dollar bills. If there are 68 bills in total, determine the quantity of twenty dollar bills.
Write a let statement defining the variable, make an equation, and solve for the unknown. Show your work.
SolutionThe number of twenty dollar bills a bank teller has is 4 less than 3 times the number of fifty dollar bills. If there are 44 bills in total, determine the quantity of fifty dollar bills.
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fifty dollar bills
Hint Unavailable
Let 'x' represent the number of 50 dollar bills.
Therefore there are 12 twenty dollar bills.
Determine the correct equation to solve the statement below.
A bank has 88 bills consisting only of five and ten dollar bills. Determine the number of five dollar bills if the total amount of money is $550.00
Money Problems
Write a let statement defining the variable, make an equation, and solve for the unknown. Show your work.
A person has 12 bills in their wallet consisting only of five and twenty dollar bills. Determine the number of five dollar bills if the total amount of money is $105.00
Solution
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five dollar bills
Hint Unavailable
Let 'x' represent the number of five dollar bills. Let (12 - x) represent the number of twenty dollar bills.
Therefore there are 9 five dollar bills.
A store sells 'Super Awesome' phones for $500 and 'Pretty Good' phones for $100. If the store sells 95 phones and makes $15,500 in a day, determine the number of Pretty Good phones sold.
Solution
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phones
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Let 'x' represent the number of Pretty Good phones. Let (95 - x) represent the number of Super Awesome phones.
Therefore 80 Pretty Good phones were sold.
Lise has $14 less than 5 times as much money as Sarah. If Lise gives Sarah $15, then Lise will have three times as much money as Sarah. Determine how much money Sarah has in the end.
Solution
$
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Let 'x' represent Sarah's money and (5x - 14) Lise's money.
After giving $15, Lise has (5x - 14 - 15) and Sarah has (x + 15).
Sarah has $52 after Lise gives her money.
Geometry Problems
A triangle has side lengths of 2x + 4, 3x - 2, and x - 10. If the total perimeter is 5x + 4, write and equation to determine the lengths of the sides of the triangle.
Determine the lengths of the sides of the triangle based on the statement below.
SolutionA triangle has side lengths of x - 4, 2x - 8, and 3x - 4. The total perimeter is x + 14.
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The length of a rectangle is 1 less than twice the width. If the perimeter of the rectangle is 70, find the length of the sides. Write a let statement defining the variable, make an equation, and solve for the unknown. Show your work.
Solution
Hint Clear Info
Length: Width:
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Let 'x' represent the width. Let (2x - 1) represent the length.
The length is 23 and the width is 12.
The area of a circle is 4 less than twice the area of a square. The area of the circle is 2 cm^{2} more than the area of the triangle. If the total area of the shapes combined is 30 cm^{2}, determine the area of the square. Write a let statement defining the variable, make an equation, and solve for the unknown. Show your work.
Solution
area =
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cm^{2}
Hint Unavailable
Let 'x' represent the area of the square. Let (2x - 4) represent the area of the circle. Let (2x - 4- 2) represent the area of the triangle.
∴ the area of the square is 8cm^{2}
Geometry Problems
Write a let statement defining the variable, make an equation, and solve for the unknown. Show your work.
A big screen TV has a display size of 160 cm x 90 cm with a bezel edge around the display that is same width all around. If the perimeter of the outer edge sides is 512 cm, determine the width of the bezel.
Solution
width =
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cm
Hint Unavailable
Let 'x' represent the width of the bezel. If the display is 160cm wide, then including the bezel it is (x + 160 + x) = (160 + 2x). The entire height is (x + 90 + x) = (90 + 2x).
∴ the width of the bezel is 1.5 cm
Write a let statement defining the variable, make an equation, and solve for the unknown. Show your work.
The high speed trains in Japan have an average speed of 240 km/hr and the high speed trains in North America have an average speed of 140 km/hr. If the train in North America leaves the station 1.0 hour before the train in Japan leaves its station, determine the time it takes the Japanese train to catch up by travelling the same distance.
Solution
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hours
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Let 'x' represent the time the Japanese train takes. Let x + 1.0 represent the time the North American train takes.
Town A is 100km apart from town B. If a car leaves town A travelling at 100km/hr at the same time that a bicycle leaves town B travelling at 15 km/hr, determine the distance travelled by the bicycle when they meet.
Solution
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km
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Let 'x' represent the distance travelled by the bicycle. Let (100 - x) represent the distance travelled by the car.
Josh is competing in an NYC race in which he can bike 10km/hr, and run 20km/hr. If each event is the same distance, and the total time is 1.5 hours, find the time it takes Josh to run.
Solution
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hour
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Let 'x' represent the time to run, and let (1.5 - x) represent the time to bike.
∴ Josh runs the event in 1.0 hours.
Linear Relations
Independent and Dependent Variable
Which of the following relationships has its independent variable underlined?
Solution
The independent variable (x-axis) is the cause or input, and the dependent variable (y-axis) is the effect, the value that depends on the input. Time is almost always an independent variable.
Pizza for a whole school of 300 people costs $3.50 per student plus a $45 delivery fee. Determine the total cost and in your own notes, determine the dependent variable, independent variable, the initial value, and the rate.
Solution
Total cost = $
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Dependent variable (the one that changes based on independent): Cost $
Independent variable: Number of People
Initial value: $45
Rate: $3.50 per student
Relations and Ordered Pairs
Which of the following ordered pairs is on the graph of y = 2x - 3?
Solution
Only one of the coordinates will work when entered into the equation given.
Plug in the x-value.
(3, 3) works.
Rates and Constants
Determine the constant in the following relation. A large pizza costs $12 plus $1 per extra topping.
Solution
The constant is also known as the initial or fixed amount.
Direct and Partial Variation
Which of the following is consistent with a direct variation relation?
Solution
A direct variation is in the form y = mx. There is no fixed cost.
A partial variation is in the form y = mx + b.
The direct variation always crosses through the y-axis at the origin (0, 0).
Which of the following equations is not a partial variation?
Solution
A direct variation is in the form y = mx.
A partial variation is in the form y = mx + b.
Determining Rates of Different Scenarios
Determine the rate for each of the following scenarios
A car can travel 550km on a 40L tank of gas. Find the rate of fuel consumption per Liter of gas.
Solution
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km/L
Hint Unavailable
A mechanical orange picker can collect 2000 oranges in 4 minutes. Find the rate of collection in seconds.
Solution
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oranges/second
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A painter can paint 336m^{2} of a building in 48 minutes. Find the rate of painting, in minutes.
Solution
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m^{2}/min
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A dog can eat 58 kernels of spilled popcorn in 26.5 seconds. Find the rate of consumption.
Solution
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kernels/sec
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Calculating Slope
Given the equations...
Calculate the slope between the points P_{1}(3, -2) and P_{2}(4, 8).
Solution
m =
Hint Clear Info
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Givens:
x_{1} = 3
y_{1} = -2
x_{2} = 4
y_{2} = 8
Note: be careful when subtracting negative 2, it becomes positive!
Calculate the slope between the points P_{1}(-4, 5) and P_{2}(2, -1).
Solution
m =
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Givens:
x_{1} = -4
y_{1} = 5
x_{2} = 2
y_{2} = -1
Note: be careful when subtracting negative 4, it becomes positive!
Rank the following slopes from lowest to highest.
Solution
Slope
Rise (cm)
Run (cm)
I
3
4
II
4
6
III
5
7
IV
2
8
Calculate the slope for each and order from lowest to highest.
Calculate the slope when the change in x is -6 with a first difference of 3.
Solution
The first difference is the .
A ladder is leaning up against a wall 4m above the ground. If the bottom of the ladder is 2m from the base of the wall, determine the slope of the ladder.
Solution
m =
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Rate of Change
A cyclist records their distance travelled as they ride their bicycle.
Time, x (hours)
Distance, y (km)
1
10
2
20
3
30
4
40
Determine the distance travelled from 1 to 4 hours.
Solution
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km
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We don't add all the distances. The cyclist starts at 10km and goes to 40km. The total distance travelled between 10km and 40km is:
means "change".
Calculate the rate of change (slope) of the distance over time from 1 to 4 hours.
Solution
m =
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km/hr
Hint Unavailable
Givens:
x_{1} = 1
y_{1} = 10
x_{2} = 4
y_{2} = 40
First (Finite) Differences
Which of the following relations is linear?
Solution
Linear relations have the same first difference.
Application of Slope
The slope of a relation for a long-distance swimming race is shown below. Determine the time it takes for a swimmer to travel a total of 800m.
Solution
t =
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s
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Writing Linear Equations
A line has a slope of 2 and a y-intercept of 3. The equation of a line in slope-y-intercept form is given below.
Write an equation for a line with a slope of that passes through the point (4, 2). Show your work and reduce fully.
Solution Video
Hint Clear Info
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Givens:
m =
x = 4
y = 2
Need to calculate b!
Write an equation for a line with a slope of that passes through the point (-1, -3). Show your work.
Solution
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Need to calculate b!
Ax + By + C = 0 and y = mx + b
The following equations are the same, where the first one is in standard form, and the second one is in slope y-intercept form.
Solutiony = ¼x - 3 -x + 4y + 12 = 0
See how to convert from slope y-intercept into standard form:
Use the slope calculate from before (m = 3), and make an equation like this:
Then choose either of the points M(2, -2) or N(5, 7) to plug into the equation to find 'b':
Put the slope (m) and y-intercept (b) from before into an equation:
Ax + By + C = 0 and y = mx + b
Rearrange the equation in standard form below, into slope y-intercept form: y = mx + b. Show your work.
Solution
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Writing Linear Equations
Which of the following relations has a slope of -2 and a y-intercept at 8?
Solution
Rearrange to check in y = mx + b form.
m = -2 b = 8
Relations
According to the following table, the average score on a test depends on the age of the student.
Solution
Age
Score
Grade 5
76%
Grade 6
77%
Grade 7
75%
Grade 8
79%
Grade 9
76%
There is no correlation (neither positive, nor negative). To determine this, see that the change in 'y' (score), for a given change in 'x' is inconsistent.
Relationship Between Two Variables
Complete the following table relating the side length of a square to its area.
Solution
Hint Clear Info
Side Length (cm)
Area (cm^{2})
2
4
5
25
7
10
144
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Side Length (cm)
Area (cm^{2})
2
4
5
25
7
49
10
100
12
144
Graphing Relations
Make a table of values (at 0km, 10km, 20km, 30km, 40km, 50km) and a graph to display the relationship of renting a car with a fixed cost of $20.00 plus a variable rate of $0.50 per kilometer. What is the cost if the car is driven 40km?
Solution
Describe the correlation of the trend on the following graph:
Solution
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Know the terms,
Strong vs. Weak
Positive vs. Negative
The trend is strong because the points appear to hover near a line of best fit. The trend is positive because as the independent axis increases in value, the dependent axis increases in value.
Graphing
Graph the data in the following table on a scatter plot. Make sure to properly label and set the dependent and independent variables on the correct axis.
Solution
Number of White Roses
Cost of Flower Arrangement
1
$3.00
4
$12.00
8
$24.00
12
$36.00
16
$48.00
24
$65.00
Make points on the axis that have equally spaced intervals, like 5, 10, 15, 20, 25, 30, etc... Don't use 1, 4, 8, 12, 16, 24 on your x-axis because those points have uneven spacing!
Make sure to label the axes and make a title of the graph.
The trend is strong because the points appear to hover near a line of best fit. The trend is positive because as the independent axis increases in value, the dependent axis increases in value.
The flight of an airplane descending to land can be modeled by the equation y = 400 - 10x, where y is height in meters, and x is time in seconds.
Find how long it will take for the plane to land on the ground.
Solution
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s
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The plane lands on the ground when height, y equals zero. Substitute y = 0 and solve for x.
∴ the plane lands on the ground after 40 seconds.
Determine the time when the plane is 150 meters off the ground.
Solution
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s
Hint Unavailable
Given:
Substitute y = 150 m into the equation
Find x
∴ the plane is 150 meters off the ground at 25 seconds.
Linear Equation Algebra
Amanda is working hard to save up to buy herself a nice iPhone from Apple. She knows it will be expensive and that money doesn't grow on trees so she wants to compare the different plan options A - D.
Plan
Phone
Cost of Phone
Cost of Plan (per month)
A
Latest iPhone
$230
$50
B
Latest iPhone
$700
$35
C
Previous Generation iPhone
$280
$50
D
Previous Generation iPhone
$450
$35
Model each of these four plans as linear equations in slope y-intercept form.
Solution
Hint Clear Info
Plan A:
Plan B:
Plan C:
Plan D:
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The cost of the phone is the fixed cost, and the monthly cost of the plan is the variable cost (slope).
Plan A...
Plan B...
Plan C...
Plan D...
Determine the lowest price plan option after 2 years of use, and its total cost.
Solution
Hint Clear Info
Plan:
Cost $
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The cost of the phone is the fixed cost, and the monthly cost of the plan is the variable cost (slope).
2 years equals 24 months, so plug 24 in for the time, x to find the lowest price, y...
Plan A...
Plan B...
Plan C...
Plan D...
Therefore Plan D is the lowest price plan option after 2 years of use.
Determine the most expensive plan option in the long term, after 3 years of use, and its total cost.
Solution
Hint Clear Info
Plan:
Cost $
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Hint Unavailable
The cost of the phone is the fixed cost, and the monthly cost of the plan is the variable cost (slope).
3 years equals 36 months, so plug 24 in for the time, x to find the lowest price, y...
Plan A...
Plan B...
Plan C...
Plan D...
Therefore Plan C is the most expensive plan option after 3 years of use.
Linear Equation Algebra
A car-rental company charges a monthly fee of $25 plus $8 per hour, n of use for a car.
Notice that this is an equation in the form y = mx + b ...
slope (m) is the hourly rate, and the y-intercept (b) is the fixed amount.
Rearrange this equation for 'n' and determine how many hours the car can be used for $465.
Solution
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hours
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C = $465
If a different car rental company charges no monthly cost with a higher rate of $12 per hour, determine which company is a better deal for the same amount of use.
Solution
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The cost is higher, $660 for the same amount of hours, so it is not a better deal.
Point of Intersection
Solve the system means find the point of intersection between two lines. If the lines do not intersect then there is no solution to the system.
Solution
'Solve the system' means find the point of intersection (POI). The solution to the system is the POI.
Which is the proper way to solve the system of equations below?
Solution
The point of intersection (POI) is (2, 1).
Determine the point of intersection in the following system...
Solution
What is a solution? When we solve a system, we get the point (coordinate). If there's no solution that means there's no coordinate - so they don't intersect.
The lines are parallel, as can be seen with the same slope, m = 2.
Lines with the same slope are parallel.
Parallel lines never cross (intersect) and have no solution (POI).
Which other line equation would have no solution in relation to ƒ(x) = -3x + 5
Solution
The other equation that has the same slope.
Come up with your own two equations that have no solutions and explain why there is no solution.
Solution
(Answers may vary: any two equations that have the same slope, m )
Linear Analytic Geometry
Deeper Understanding of Slope
Which of the following is an equation of a vertical line?
Solution
A vertical line crosses the x-axis and has no y-values.
Which of the following is an equation of a horizontal line?
Solution
Another way to think about it is a line with a slope (M) of 0, and the y-intercept (b) is the whole line.
Write the equation of a line with the same slope, that has a y-intercept 5 points higher than in part 'a'.
Solution
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Hint Unavailable
If the original y-intercept was -3.
A y-intercept 5 points higher = -3 + 5, would be cross at +2.
Solving Equations
A math student made a mistake when rearranging the equation below to find slope. Which step has a mistake?
Solution
Rearranging from Standard Form
Answer the questions for the equation:
The slope (m) is 3, and the y-intercept (b) is 2.
Solution
First you need to rearrange into slope-y-intercept form to determine this...
The slope (m) is and the y-intercept (b) is .
Determine the slope and y-intercept for the equation.
Solution
Hint Clear Info
Slope: ━━ y-intercept:━━
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To find the slope (m) and y-intercept (b), rearrange from standard form into slope-y-intercept form: y = mx + b
From the equation in the form y = mx + b, you can see that:
slope, m =
y-intercept, b =
Slope, Intercept, Points, and the Equation
Which of the following equations contains the point (4, 6) and has a slope of 8?
Solution
Substitute slope (m) and the point into the equation and solve for 'b'.
Given the slope of a line is , and the points (n, 8) and (3, 4). Determine the value of n.
Solution
n =
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Hint Unavailable
Givens:
x_{1} = n
y_{1} = 8
x_{2} = 3
y_{2} = 4
Determine the equation of a line that has the same slope as the equation below and passes through the point (2, 3)
Solution y = -2x - 5
Hint Clear Info
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Hint Unavailable
Givens:
x = 2
y = 3
m = -2
Combining Slope and Intercept in Equations
Determine the equation of a line that passes through the points below.
Solution (2, 4) and (3, -6)
Hint Clear Info
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Hint Unavailable
Find the slope, m using the points:
(2, 4)
(3, -6)
Solve for 'b' by substituting a point into the equation. Use (2, 4).
First convert to slope-y-intercept form
Then determine the negative reciprocal of the slope.
Switch the signs, and flip (reciprocate) the number or fraction...
New y-intercept calculation not shown.
Parallel and Perpendicular Lines
Determine the equation of a line that is perpendicular to the line 2x - 4y = 2 and has the same y-intercept as the line x + 7y = 4.
Solution Video
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Hint Unavailable
Perpendicular tells us the slopes are negative reciprocals (flip the number, and change the signs). So find the slope of the line in slope-y-intercept form, and then flip and switch the slope: 2x - 4y = 2
The slope is 1/2, so the negative reciprocal is: m = -2/1 = -2.
Find the y-intercept of the line x + 7y = 4
The y-intercept is 4/7, then putting this together with m = -2...
Zero and Undefined Slope
A horizontal line has a slope of zero and a y-intercept of +4. Determine the equation of the line.
Solution
y =
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Slope of zero: m = 0...
Horizontal lines are in the form y = constant ...
Complete the table of coordinates using the line equation determined previously.
Solution
Hint Clear Info
x
y
-2
-1
0
1
2
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Hint Unavailable
For all the points on the horizontal line, y = 4...
x
y
-2
4
-1
4
0
4
1
4
2
4
Zero and Undefined Slope
A vertical line passes through the points (-5, 10) and (-5, 0). Determine the slope.
Solution
Hint Clear Info
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Hint Unavailable
Slope (m) equals...
The denominator zero cannot be calculated and is infinite/undefined. We say the slope of vertical lines is undefined.
Since the slope (m) is undefined, we do not have a value for m. At all points, everywhere...
Collinear
Three points A, B, and C are considered collinear if they are on the same line.
Solution
True. Collinear means lying on the same straight line.
The following points are collinear. Solution A(-8, -3), B(1, 1), C(5, 4)
The points are not collinear because they do not lie on the same line, because they do not have the same slopes.
Determine the value of n that would make the following points collinear. Solution (-2, -10), (3, 5), (n, -19)
n =
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Collinear points lie on the same line. Determine the slope between (-2, -10) and (3, 5).
Sub the slope into the equation with another point.
Solving Geometric Equations
A gallery has 160cm of picture framing wood left to use. Write an equation for the different dimensions of picture frames that can be made. Make 'y' the width and 'x' the length. Using your equation, determine which of the following is not a possible combination. Show your work.
Solution
It should be: 30 cm length, 50 cm width.
Linear Analytic Equations
Wholesale T-shirts cost $8 for organic cotton and $6 for cotton blend.
If Sophie paid $442 for her wholesale T-shirt order, write an equation to represent the total cost for the order. Let n represent number of organic shirts, and let m represent the number of blend shirts.
Solution
cost =
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Hint Unavailable
If Sophie bought 23 cotton blend T-shirts, rearrange the equation and solve for the number of organic shirts purchased.
Solution
n =
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shirts
Hint Unavailable
Let n represent number of organic shirts, and let m represent the number of blend shirts.
Measurement and Geometry
Classifying Shapes
A rhombus is a parallelogram with equal side lengths, including squares and rectangles.
Solution
A rhombus is a parallelogram with equal side lengths.
Angles on a straight line:
Sum of the angles in a triangle:
Put 'B' from equation 2 into equation 1. (Make sure to distribute the negative into the brackets.)
Interior Angle Algebra
The interior angles of a triangle are given in terms of x in the diagram below.
A median extends from one vertex to the midpoint of the opposite side. (A median can be, but does not have to be, perpendicular to the side length.)
Volume of Prisms
The formula shown is used to calculated the volume of:
SolutionV = (Area of base) × (Height)
The volume of a cube, rectangular prism, cylinder, and triangular prism are calculated with the formula:
Volume
The volume of a pyramid can be calculated with the following formula:
SolutionV = (Area of base) × (Height)
This formula only works for shapes with consistent areas: cube, cylinder, triangular prism, rectangular prism.
This does not work for pyramids, cones, or spheres.
Volume of Prisms
Determine the volume of the triangular prism below, where the height of the triangle is 5 cm, the base is 4 cm, and the height of the prism is 10cm.
Solution
Hint Clear Info
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cm^{3}
Hint Unavailable
Volume = (Area of Base) × (Height)
Volume
A cylinder contains three spheres stacked tightly, which touch the top, sides, and base of the cylinder. The circumference of the sphere and cylinder are equal.
If the circumference is 8 cm, determine the radius.
Solution
r =
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cm
Hint Unavailable
Determine the volume of the three spheres. Round your answer to the nearest whole number.
Solution
V =
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cm^{3}
Hint Unavailable
It is impossible to determine the height of the cylinder with the given information.
Solution
Since the spheres area, "stacked tightly, which touch the top, sides, and base of the cylinder" then the height is a multiple of the radius, r.
From looking at the diagram, you can see the height = 6 × r
Determine the volume of the space between the three spheres and the inside of the cylinder. Round your answer to the nearest whole number.
Solution
V =
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cm^{3}
Hint Unavailable
The volume of the 3 spheres, calculated previously (not shown), = 804 cm^{3}.
Height of cylinder = 3 sphere = 6 radius' = 4 × 6 = 24 cm.
Perimeter and Area
The quadrilateral with the ratio of lowest perimeter to highest area is the rectangle.
Solution
The quadrilateral with the ratio of lowest perimeter and highest area is the square.
Perimeter and Area
A pizza place sells 8" (diameter), 12", and 16" pizzas. How many times larger is the area of the 16" than the 8" pizza?
Solution
Hint Clear Info
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times larger
Hint Unavailable
Remember this is area... proportional to the square of the radius. Doubling the radius or diameter is always equivalent to quadrupling (4×) the area, which is the amount of pizza that you get.
For example,
Hopefully you can see that the area of the largest pizza is 4x the area of the smallest pizza,
Perimeter and Area
A circle and rectangle have the same area (not drawn to scale). The radius of the circle is 3cm, and the width of the rectangle is 6cm.
Determine the length of the rectangle, round your answer to one decimal place.
Solution
length =
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Hint Unavailable
The statement says the circle and rectangle have the same area:
Determine which shape has a larger perimeter, show your work.
Solution
Therefore the rectangle has a larger perimeter.
Area of Compound Shapes
This large Roman window arch is made of a semicircle (half a circle) and a rectangle. The base (b) of the window is 2.0 m and the total height, h is 4.0 m
Since the base is the same length as the diameter of the semicircle, then the radius is 1.0 m. The area of the semicircle is half a circle.
Calculate the rest of the area to determine the total area of the window.
Solution
A =
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m^{2}
Hint Unavailable
See that the total height includes the radius (height) of the semicircle of 1.0 m
This means the height of the rectangle is 3.0 m
Surface Area Cylinder
The surface area of the shape below can be calculated with which of the following formulas?
Solution
The top and bottom of the cylinder are circles. The area for a circle is . There are two of them, . The circumference of the circle matches the side length of the rectangle, so that is the "width" of the rectangle. The area of the rectangle is (width)(height) = .
is the formula for surface area of a cylinder. Think of the sum of the compound shapes.
The surface area of a cylinder is 226.2cm^{2}, and the radius is 4cm. Determine the height, showing your work.
Solution
h =
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cm
Hint Unavailable
Optimization of Rectangle Area
A pet store owner has 90cm of fencing to enclose a baby turtle with three walls of fencing. Determine the dimensions of fencing that would allow for a maximum area of the enclosure for the baby turtle. Complete the table in your notebook, and show your work.
Solution
Length (cm)
Width (cm)
Area (cm^{2})
1
___
___
2
___
___
3
___
___
5
___
___
6
___
___
9
___
___
10
___
___
15
___
___
18
___
___
30
___
___
45
___
___
The 30cm × 30cm = 900cm^{2}.
By inspection, you see that the 30cm × 30cm enclosure has the maximum area, 900cm^{2}.
Area of Cutouts
Two equal circles overlap a square with a side length of 10cm. Determine the shaded area.
Solution
A =
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cm^{2}
Hint Unavailable
If length of square = 10cm, then the radius of the circle = 5cm.
As you can see in the diagram, the area of the square on the longest side (hypotenuse) is 5^{2} = 25. The sum of the areas of the squares on the two shorter sides is 3^{2} + 4^{2} = 9 + 16 = 25.
∴ The area of the larger square is always equal to the sum of the two smaller areas.
Pythagorean Theorem
Which of the following is the correct formula to solve for side length b in a right triangle using the Pythagorean theorem?
Solution
Pythagorean Theorem
Two right triangles share a side length 'c' in the diagram below.
Determine the length of the hypotenuse 'c'.
Solution
The diameter of a cylinder is 4x and the height, h is 2x. Determine the surface area in terms of x and .
Solution
Diameter of a cylinder is 4x, therefore the radius, r = 2x.
Surface Area
A cone has a surface area of 400 cm^{2}. If the radius is 5 cm, determine the slant height (L), round your answer to 1 decimal place.
Solution
L =
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cm
Hint Unavailable
Solve for L.
Be careful with order of operations (BEDMAS).
Surface Area
A cylinder has half a sphere (dome) placed on top if it and shares the same radius, 4cm. The height of the cylinder is 6cm.
The total surface area of the shape is equal to the surface area of the cylinder plus the surface area of half a sphere, according to the equation below.
Solution
The dome is in contact with the top of the cylinder. The top of the cylinder is not included in the calculation of the total surface area.
Determine the total surface area of the shape.
Solution
SA =
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cm^{2}
Hint Unavailable
One of the circles on the cylinder is covered by the dome, so cancel one of the circle areas in the equation below,
Volume and Surface Area
A cone is attached to a cylinder with a radius of 12 cm, shown below. The height of the cone is 16 cm, and the height of the cylinder is 30 cm.
Determine the slant height (L) of the cone.
Solution
L =
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cm
Hint Unavailable
The slant height of the cone is the longest side (hypotenuse) of the right angle triangle formed with the height and radius of the cone.
Therefore the slant height is 20 cm.
Calculate the volume of the whole shape given the equations below.
Solution
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cm^{3}
Hint Unavailable
Determine the surface area of the outside of the shape.
Solution
SA =
Hint Clear Info
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cm^{2}
Hint Unavailable
L = 20 cm
BE CAREFUL NOT TO USE THE FULL SURFACE AREA OF THE CONE AND CYLINDER!
Do not include the base of the cone:
Do not include the top of the cylinder:
Optimization of 3D Shapes Given Volume
A square-based prism has a surface area equal to 96 cm^{2}.
The equation for the volume of the figure, in terms of b and h, equals b^{2}h.
Solution
The volume of a square-based prism is the area of the base (b^{2}) × height.
The equation for the surface area of the figure, in terms of b and h, equals 6bh.
Solution
Determine an equation for height, h in terms of b.
Solution
Determine the maximum volume of this square-based prism, and the dimensions of the base and height.
Solution
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cm^{3}
Hint Unavailable
Base (cm)
Height (cm)
Volume (cm^{3}) = b^{2} × h
1
23.5
= 1^{2} × (23.5) = 23.5
2
11
= 2^{2} × (11) = 44
3
6.5
= 3^{2} × (6.5) = 58.5
4
4
= 4^{2} × (4) = 64
5
2.3
= 5^{2} × (2.3) = 57.5
6
1
= 6^{2} × (1) = 36
Therefore the maximum volume (64 cm^{3}) occurs when the base equals 4cm, and the height equals 4 cm.