Question

The area of a square is given by the function $a(x)=x^{2},$ where $x$ is the length of the sides of the square. If carpeting costs $\$ 9.95$ per square yard, then the cost, $C$, to carpet a square room is given by $C(a)=9.95 a$ where $a$ is the area of the room in square yards. a. Find $(C \circ a)(x)$ and interpret its meaning in the context of this problem. b. Find the cost to carpet a square room if its floor dimensions are 15 yd by 15 yd.

Name: Problem 86, Chapter 8: Beginning and Intermediate Algebra
Uploaded: 2021-08-02T23:24:28-08:00
Duration: 2 min 24 s
Channel: Andrew Lebedinsky

   The area of a square is given by the function $a(x)=x^{2},$ where $x$ is the length of the sides of the square. If carpeting costs $\$ 9.95$ per square yard, then the cost, $C$, to carpet a square room is given by $C(a)=9.95 a$ where $a$ is the area of the room in square yards. a. Find $(C \circ a)(x)$ and interpret its meaning in the context of this problem. b. Find the cost to carpet a square room if its floor dimensions are 15 yd by 15 yd.

Beginning and Intermediate Algebra

Julie Miller,Molly… 3rd Edition

Chapter 8, Problem 86

Instant Answer

Solved by Expert Andrew Lebedinsky

08/02/2021

Step 1

95a$. We are asked to find the composition of these functions, which is denoted as $(C \circ a)(x)$. This means we substitute $a(x)$ into $C(a)$. Show more…

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The area of a square is given by the function $a(x)=x^{2},$ where $x$ is the length of the sides of the square. If carpeting costs $\$ 9.95$ per square yard, then the cost, $C$, to carpet a square room is given by $C(a)=9.95 a$ where $a$ is the area of the room in square yards. a. Find $(C \circ a)(x)$ and interpret its meaning in the context of this problem. b. Find the cost to carpet a square room if its floor dimensions are 15 yd by 15 yd.

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Key Concepts

Composite Functions

Composite functions involve applying one function to the results of another. In this scenario, the cost function is composed with the area function, which means you first calculate the area of the room using the function that squares the side length, and then calculate the cost by applying the cost per unit area to that area value.

Area Calculation of a Square

The area of a square is determined by squaring the length of one of its sides. This principle is used to calculate how much floor space there is to cover, by using the function a(x) = x², where x represents the length of a side.

Cost Calculation per Unit Area

When a cost is given per square yard, multiplying this cost by the total area yields the overall cost. The cost function, in this case C(a) = 9.95a, uses the calculated area to determine how much carpeting is needed based on the given cost per square yard.

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