The ice skating rink charges an hourly fee for skating and $3 to rent skates for the day. Gillian rented skates and skated for 3 hours and was charged $21. Which equation represents the cost, $c(x)$, of ice skating as a function of $x$, the number of hours of skating?
A. $c(x)=3x+3$
B. $c(x)=6x+3$
C. $c(x)=7x+3$
D. $c(x)=8x+3
Answer (2)
Define the hourly fee for skating as h , and set up the equation 3 h + 3 = 21 based on the given information.
Solve the equation for h to find the hourly fee: h = 6 .
Construct the cost function c ( x ) as a function of the number of hours x : c ( x ) = 6 x + 3 .
The equation representing the cost of ice skating is c ( x ) = 6 x + 3 .
Explanation
Problem Analysis Let's analyze the problem. We know that the ice skating rink charges an hourly fee for skating, plus a fixed fee for renting skates. Gillian skated for 3 hours and rented skates, and her total charge was 21. W e n ee d t o d e t er min e t h ee q u a t i o n t ha t re p rese n t s t h e t o t a l cos t , c(x) , a s a f u n c t i o n o f t h e n u mb ero f h o u rs , x$, of skating.
Setting up the Equation Let h be the hourly fee for skating. The cost of renting skates is a fixed fee of $3. Gillian skated for 3 hours, so the cost for skating is 3 h . Since she also rented skates, the total cost is 3 h + 3 . We know that the total cost was 21 , so w ec an w r i t e t h ee q u a t i o n : 3 h + 3 = 21 $
Solving for the Hourly Fee Now, let's solve for h . First, subtract 3 from both sides of the equation: 3 h = 21 − 3 3 h = 18 Next, divide both sides by 3: h = 3 18 h = 6 So, the hourly fee for skating is $6.
Finding the Cost Function Now that we know the hourly fee is 6 , w ec an w r i t e t h ee q u a t i o n f or t h e t o t a l cos t , c(x) , a s a f u n c t i o n o f t h e n u mb ero f h o u rs , x , o f s ka t in g . T h ecos t f ors ka t in g x$ hours is 6 x , and the cost of renting skates is 3. T h ere f ore , t h e t o t a l cos t i s : c ( x ) = 6 x + 3 $
Final Answer The equation that represents the cost of ice skating as a function of x , the number of hours of skating, is c ( x ) = 6 x + 3 .
Examples
Imagine you're planning a birthday party at the ice skating rink. Knowing the cost function c ( x ) = 6 x + 3 helps you calculate the total expense based on how long the party lasts. For instance, if the party lasts for 2 hours, the cost would be $c(2) = 6(2) + 3 = $15. This kind of calculation is useful for budgeting and making informed decisions about the duration of the event, ensuring you stay within your planned expenses. Understanding cost functions can be applied to various real-life scenarios, from planning events to managing personal finances.
The cost function for ice skating, based on Gillian's experience, is c ( x ) = 6 x + 3 , where $6 is the hourly fee and $3 is for skate rental. This corresponds to option B. Thus, the cost increases with each additional hour of skating.
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