Lift Lines and Snowfall Rates

Find the constant of proportionality in resort records and models.

Read the Records

For each proportional relationship, find k in y = kx. Pay attention to the order of the quantities.

A lift operator records the number of riders boarding during a steady loading period. Let x be elapsed minutes and y be riders.

Riders Boarding
Minutes, x	Riders, y
2	36
5	90
8	144

What is k? riders per minute

During one steady snowfall, x is elapsed hours and y is new snow depth in inches. Which value is the constant of proportionality y/x?

New Snow
Hours, x	Inches, y
2	1.5
4	3
6	4.5

A. 2 inches per hourB. 0.75 inch per hourC. 1.5 inches per hourD. 1 1/3 inches per hour

Each full chair on a lift carries the same number of riders. Read the loading diagram.
3 full chairs → 12 riders
7 full chairs → 28 riders
10 full chairs → 40 riders

If x is full chairs and y is riders, what is k? riders per full chair

A snow sensor records a steady 5 inches of new snow over 4 hours. New snow depth is proportional to elapsed time.

What is the constant of proportionality in inches per hour? inches per hour

A lift line has no new arrivals. At a steady boarding rate, 63 waiting riders board in 7 minutes. Let x be riders boarded and y be minutes elapsed. What is k in y = kx?

A. k = 1/9 minute per riderB. k = 1/63 minute per riderC. k = 9 minutes per riderD. k = 7 minutes per rider

Read the Models

Use the axes and variable definitions to identify each constant.

The graph models new snow depth during a steady storm. The line passes through the origin.

What is the constant of proportionality in centimeters per hour? cm per hour

This graph models the number of full lift chairs needed for a group. What is the constant of proportionality y/x?

A. 0.25 chair per riderB. 2 chairs per riderC. 4 chairs per riderD. 0.5 chair per rider

The equation y = 24x models riders boarding a lift. Here x is elapsed minutes and y is riders boarded.

What is k? riders per minute

A snowfall model is 4y = 7x, where x is elapsed hours and y is new snow depth in centimeters. What is k in y = kx?

A. k = 4/7 centimeter per hourB. k = 28 centimeters per hourC. k = 7/4 centimeters per hourD. k = 7 centimeters per hour

10.

A resort model uses y = x/6, where x is the number of riders and y is the number of full six-person chairs needed.

What is k in y = kx? chair per rider

Check the Rate

Apply the models to resort decisions. Explain your reasoning in the final question.

11.

A lift supervisor wants a boarding rate greater than 20 riders per minute. Each lift has a constant boarding rate. Which lift meets the goal?

A. Summit Lift boards 132 riders in 6 minutes.B. Pine Lift boards 96 riders in 6 minutes.C. Ridge Lift boards 140 riders in 7 minutes.D. Valley Lift boards 171 riders in 9 minutes.

12.

The diagram shows equal-length recording intervals during a steady snowfall.
1 interval → 0.4 inch of new snow
3 intervals → 1.2 inches of new snow
5 intervals → 2 inches of new snow
Each interval lasts 20 minutes. A report needs a rate per hour, not per interval.

What is the constant of proportionality in inches per hour? inches per hour

13.

A report lists elapsed time as a function of new snow depth. The relationship is proportional.

Storm Timing
Snow depth (in.), x	Elapsed hours, y
1.5	2
3	4
6	8

What is k in y = kx? hours per inch

14.

During a steady snowfall, the graph of new snow depth y in inches against elapsed time x in hours includes the point (4, 6). Nia says, “The constant of proportionality is 6 because 6 is the y-coordinate.” Explain why her method is incorrect and state the correct constant with units.