4.08 Proportional relationships in the coordinate plane
Consider the following graph:
Complete the ratio table by using the points shown on the graph.
| x | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| y |
For each of the following graphs, describe what the plotted point represents:
The graph shows the price for playing arcade games.
The graph shows the number of eggs a farmer's chickens lay each day.
The graph shows the amount of time it takes Kate to make beaded bracelets.
The graph shows the number of liters of gas used by a fighter jet per second.
The graph shows the number of liters of ice cream per tub.
The graph shows the distance Natalia swam per minute.
The number of cupcakes eaten by guests at a party is shown on the graph:
State what the point on the graph represents.
Danielle is having a party and expects to have 10 guests. According to the rate shown on the graph, how many cupcakes should she buy?
The number of fish caught by Harry last weekend is shown on the graph:
State what the point on the graph represents.
How many fish would Harry expect to catch in 1 hour?
The amount of chocolate and milk that can be heated up together to make a hot chocolate is shown on the graph:
State what the point on the graph represents.
How many spoons of chocolate should be mixed with 1 cup of milk?
The following graph shows the proportion of red sweets and green sweets in a mix:
Find the number of red sweets for every 1 green sweet in a mix.
The following graph shows the number of patients that are seen over time:
What ratio has been plotted?
Write a statement to describe the number of patients being seen evey hour.
The following graph shows the distance traveled over time:
Describe how the distance traveled changes over time.
State the ratio of distance to time.
The following graph shows the ratio of the number of cups of strawberries to the number of cups of raspberries required in a recipe:
What ratio has been plotted?
Describe the relationship between the number of cups of strawberries and the number of cups of raspberries required.
The ratio of x:y is 2:6. By finding two points that represent equivalent ratios to 2:6, graph the ratio on a coordinate plane.
For each of the following ratios of x:y, plot the points from the table of values on a coordinate plane:
The ratio of x:y is 1:2.
| x | 3 | 4 | 5 | 6 |
|---|---|---|---|---|
| y | 6 | 8 | 10 | 12 |
The ratio of x:y is 1:1.
| x | 3 | 4 | 5 | 6 |
|---|---|---|---|---|
| y | 3 | 4 | 5 | 6 |
For each of the following proportional relationships:
Complete the table of values.
Plot the points from the table of values.
Draw the graph of the proportional relationship between x and y.
The ratio of x:y is 1:3.
| x | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| y |
The ratio of x:y is 4:1.
| x | 4 | 8 | 12 | 16 |
|---|---|---|---|---|
| y |
The cost of parking for various amounts of time was recorded at 4 different locations in the city. The following table summarizes the results:
| \text{Number of hours} \left(x\right) | 3 | 3.5 | 12 | 8 |
|---|---|---|---|---|
| \text{Cost of parking} \left(y\right) | \$13.50 | \$16.75 | \$54.00 | \$34.50 |
Find the cost per hour at the location where parking cost \$13.50.
Is x proportional to y? Explain why this might be.
While playing a particular song, a pianist is measured to make an average of 40 keystrokes every 8 seconds.
Complete the table of values:
| Time (in seconds) | 0 | 4 | 8 | 16 | 24 |
|---|---|---|---|---|---|
| Keystroke |
Graph the linear relationship represented in the table.
Find the number of keystrokes the pianist is making per second.
A diver starts at the surface of the water and starts to descend below the surface at a constant rate. The table shows the depth of the diver during the first 4 minutes:
| Number of minutes passed | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| Depth of diver in meters | 0 | 0.8 | 1.6 | 2.4 | 3.2 |
Graph the linear relationship represented in the table.
Calculate the increase in depth each minute.
Calculate the depth of the diver after 24 minutes.
Gas costs a certain amount per gallon. The table shows the cost of various amounts of gas in dollars:
| \text{Number of gallons } (x\text{)} | 0 | 10 | 20 | 30 | 40 |
|---|---|---|---|---|---|
| \text{Cost of gas } (y\text{)} | 0 | 24.00 | 48.00 | 72.00 | 96.00 |
Graph the linear relationship represented in the table.
How much does gas cost per gallon?
How much would 70 gallons of gas cost at this unit price?
Consider the following table that shows the temperature of a metal plate, in \degree\text{F}, after an amount of time, measured in minutes:
| \text{Time }(x) | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| \text{Temperature }(y) | 50 | 75 | 100 | 125 | 150 |
Graph the linear relationship represented in the table.
By how much is the temperature increasing each minute?
Find the temperature at time 0.
After Sally starts running, her heartbeat increases at a constant rate as shown in the table:
| \text{Number of minutes passed } (x\text{)} | 0 | 2 | 4 | 6 | 8 | 10 | 11 |
|---|---|---|---|---|---|---|---|
| \text{Heart rate } (y\text{)} | 75 | 81 | 87 | 93 | 99 | 105 |
Complete the table.
Graph the linear relationship represented in the table.
By how much is her heartbeat increasing each minute?
Write an equation of the form y=kx that represents each of the following proportional relationships:
Frank serves 8 cups of coffee every 9 minutes. Use y for the number of cups of coffee, and x for the amount of minutes that have passed.
For every 6 minutes that Emma runs, she covers 850 meters. Use y for the distance Emma runs in x minutes.
The amount of yellow and red paint needed to make 'sunset orange' is shown on the graph:
Let x represent the amount of yellow paint and y represent the amount of red paint needed. Determine the equation of this line.
The number of batches of cookies, y, that can be made in a bakery every hour, x, is shown in the graph:
Determine the equation of this line.
The amount of white and red paint needed to make 'flamingo pink' is shown in the graph:
Let x represent the amount of white paint and y represent the amount of red paint needed. Determine the equation of this line.
Describe the relationship between the number of cans of red paint and white paint needed.
The original of a printed image measures 8.5 centimeters in width and 34 centimeters in length. When a customer wants to print a copy of this original they are offered prints in various sizes, but the width and length are in the same ratio as the original so that the photo does not appear distorted.
If x represents the width and y represents the length of the printing size, write an equation relating x and y.
Find the length of a copy if the width of the copy is 13 \text{ cm}.
During a group gym session, participants need to do "double unders" skipping for 10 minutes. Those who can’t do double unders are given the option of doing single skips, but they must do more. The table shows the pattern of how many double under and single skips participants had to do for the first 4 minutes.
| \text{Time} \left(t \text{ minutes}\right) | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| \text{Double under skips} \left(x\right) | 19 | 38 | 57 | 76 |
| \text{Single skips} \left(y\right) | 76 | 152 | 228 | 304 |
Is the number of double under skips proportional to the number of minutes?
Write an equation for x, the number of double under skips done in t minutes.
How many double under skips did participants do in 10 minutes?
For participants who did single skips, how many single skips counted as 1 double under skip?
Write an equation relating x, the number of double under skips, to y, the number of single skips.
How many single skips did participants have to do in 10 minutes?
A car dealership has received 88 of the latest model Subaru. The dealership hopes to have sold all of them within 11 weeks. The number sold by the end of the second and fourth weeks is shown in the table:
| \text{Week} \left(w\right) | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| \text{Number of cars sold} \left(n\right) | 12 | 24 |
Assuming the new Subarus are being sold at a constant rate, complete the table.
Write an equation relating w, the number of weeks passed, and n, the number of cars sold.
If the new models continue to be sold at this rate, how many will be sold in 11 weeks?
How many of the new models will remain after 11 weeks?
How many new models would they have had to sell each week to have had them all sold in 11 weeks?
Heavy rainfall has resulted in flooding of some roads. The water level has been monitored on a particular road as shown in the table:
| \text{Number of hours} \left(t\right) | 0 | 0.5 | 1 | 1.5 | 2 |
|---|---|---|---|---|---|
| \text{Water level in inches} \left(x\right) | 0 | 0.75 | 1.5 | 2.25 | 3 |
Is the water level proportional to the number of hours passed?
Write an equation relating x and t.
After 2 hours of rainfall, authorities make a decision to close the road when the water level reaches 7.5 \text{ in}. If it continues to rain at the same rate, how much longer do residents have to access the road before it closes?