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Chapter 1: Problem 68

A rectangle is bounded by the \(x\) -axis and the semicircle \(y=\sqrt{36-x^{2}}\) (see figure). Write the area \(A\) of the rectangle as a function of \(x,\) and graphically determine the domain of the function.

Short Answer

Expert verified
The area \(A\) of the rectangle as a function of \(x\) is \(A = 2x \times \sqrt{36-x^{2}}\), and the domain of the function is \(-6 \leq x \leq 6\).

Step by step solution

01

Understand multiple components

We understand a rectangle's area as \(Area = length \times width\). Our rectangle's horizontal side (\(length\)) is \(2x\), and the vertical side (\(width\)) is \(y\). The vertical side is along the semicircle with equation \(y = \sqrt{36-x^{2}}\). We will express the area of the rectangle in terms of \(x\).
02

Write the area as a function of \(x\)

Replace \(width = y\) in the area equation with the semicircle function \(y = \sqrt{36-x^{2}}\) we have: \(Area = length \times width = 2x \times \sqrt{36-x^{2}}\)
03

Determine the domain of the function

The domain of the function is the set of all valid values of \(x\) for which the function is defined. Because we're dealing with a semicircle, we know that \(x\) must be within the radius of the semicircle which is \(6\). Therefore, the function is only defined for \( -6 \leq x \leq 6 \).

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