A company is designing a circular logo for a product. The inner part of the logo is a circle with a radius of r inches. The logo has a boulder that is a concentric circle surrounding the inner circle. The area of the border is 14.5 square inches. The logo is shown. Write an expression that represents the total area, in square inches, of the logo in terms of the radius, r, of the inner part of the logo. If the total area of the logo is 127 square inches, what is the approximate value of r? Show your work or explain your work.

a. A' = πr² + 14.5

b. 5.98 in

Step-by-step explanation:

a. Write an expression that represents the total area, in square inches, of the logo in terms of the radius, r, of the inner part of the logo.

Since the inner circle has a radius of r, its area is A = πr².

Now, the border has and area of 14.5 in². Thus, the total area, A' = area of inner circle + area of border

A' = πr² + 14.5

b. If the total area of the logo is 127 square inches, what is the approximate value of r?

Given that A' = total area of logo = 127 in²

So, A' = πr² + 14.5

πr² = A' - 14.5

dividing through by π, we have

r² = (A' - 14.5)/π

taking square-root of both sides, we have

r = √[(A' - 14.5)/π]

Substituting the value of A' into the equation, we have

r = √[(127 in² - 14.5 in²)/π]

r = √[(112.5 in²/π]

r = √[35.81 in²]

r = 5.98 in

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