The area of a square is given by the function a(x)=x^2,...

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 $C(a)$ (in dollars) 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.

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 $C(a)$ (in dollars) 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

Function Composition

Function composition involves applying one function to the result of another. In this context, the cost function is composed with the area function by substituting the area of the square (given by squaring the side length) into the cost function, thereby linking the side length directly to the cost.

Area of a Square (Quadratic Function)

The area of a square is calculated by squaring the length of its side. This relationship is represented by a quadratic function, which highlights how the area grows as the square of the side length. It is a fundamental geometric principle used to relate linear dimensions to two-dimensional space.

Cost Function Based on Area

A cost function converts a measurable quantity (here, area) into a cost by applying a rate (dollars per square yard). By multiplying the area by the cost per unit area, the function provides the total expense for carpeting a room, making it a practical application of algebra in real-world scenarios.

Evaluating Functions

Evaluating functions means substituting specific numerical values into a function to obtain a particular output. In this problem, it involves calculating the cost of carpeting by inputting a given side length into the composed function, thereby obtaining an explicit cost value for a room of that size.

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