Question

1. shawn and her friends love to go ice skating. last week, shawn went to the ice rink and paid $15 to skate for 2 hours. today, her friend sara skated for 3 hours at the same ice rink and paid a total of $22.50.

a. suppose the relationship between the total cost and the number of hours of skating is proportional. complete the table with two additional pairs of values. explain what these values mean in terms of a constant rate.

number of hours of skating	total cost (dollars)
3	22.50

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b. suppose the relationship between the total cost and the number of hours of skating is not proportional. complete the table with two additional pairs of values. explain what these values might mean in terms of a constant rate.

number of hours of skating	total cost (dollars)
3	22.50

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Explanation:

Step1: Find the constant rate for proportional case

If the relationship is proportional, the rate $r$ (cost per hour) is found by dividing the cost by the number of hours. For the first - given pair, $r=\frac{15}{2}=7.5$ dollars per hour.

Step2: Generate new pairs for proportional case

For 4 hours, the cost $C = 7.5\times4=30$ dollars. For 5 hours, the cost $C = 7.5\times5 = 37.5$ dollars.

Step3: For non - proportional case

Let's assume a non - proportional relationship with an initial fee $F$ and an hourly rate $h$. We have the system of equations: $15=F + 2h$ and $22.5=F + 3h$. Subtract the first equation from the second: $(F + 3h)-(F + 2h)=22.5 - 15$, so $h = 7.5$ and $F=0$ (in this case it still seems proportional, but let's assume a different structure). Let's say there is a one - time equipment fee of $5$ dollars and an hourly rate of $5$ dollars.
For 4 hours: $C=5+5\times4=25$ dollars. For 5 hours: $C=5 + 5\times5=30$ dollars.

a.

Number of Hours of Skating	Total Cost (dollars)
5	37.50

The constant rate is $7.5$ dollars per hour. This means for every additional hour of skating, the cost increases by $7.5$ dollars.

b.

Number of Hours of Skating	Total Cost (dollars)
5	30.00

Here, we assumed a non - proportional relationship with a one - time equipment fee of $5$ dollars and an hourly rate of $5$ dollars. The cost increases by $5$ dollars per additional hour after paying the initial $5$ dollar equipment fee.

Answer:

a. 4, 30.00; 5, 37.50
b. 4, 25.00; 5, 30.00