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Question

# The area A of a circle in terms of the circle's radius r is A= πr². a. Explain the restrictions on the domain of the function A = πr² and the inverse function in this context. b. Write the inverse function of A = πr². c. Use the inverse function you wrote in part (b) to find the radius of a circle that has an area of 50.25 in.². Round your answer to the nearest inch.

Solution

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$\textbf{(a)}$

A negative radius does not make sense so we restrict the radius to:

$\color{#c34632}{r\geq 0}$

$\textbf{(b)}$

To find the inverse of $A$, we simply solve for $r$ in terms of $A$. Dividing both sides by $\pi$, we have:

$\dfrac{A}{pi}=r^2$

Take the positive square root of both sides:

$\sqrt{\dfrac{A}{\pi}}=r$

$\color{#c34632}{r=\sqrt{\dfrac{A}{\pi}}}$

$\textbf{(c)}$

Using the inverse function from part (b), substitute $A=50.25$ and solve for $r$:

$r=\sqrt{\dfrac{50.25}{\pi}}$

$\color{#c34632}{r\approx 4\text{ in.}}$

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