Exercise 50 Page 838

Let's analyze the given diagram.

We are asked to find the ratio of the area of the circle to the area of the square. The radius of the circle is r=2x. Now we can find its area.

\textcolor{darkorange}{A_\text{circle}}=\pi r^2

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Substitute 2x for r and evaluate

Substitute

r= 2x

\textcolor{darkorange}{A_\text{circle}}=\pi ({\color{#0000FF}{2x}})^2

PowProd

(a * b)^m=a^m* b^m

\textcolor{darkorange}{A_\text{circle}}=\pi (2^2)x^2

CalcPowProd

Calculate power and product

\textcolor{darkorange}{A_\text{circle}}=\textcolor{darkorange}{4\pi x^2}

The side length of the square s is twice the radius of the circle. This tells us that s=2(2x)=4x. Now, let's find the area of the square.

\textcolor{darkviolet}{A_\text{square}} = s^2

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Substitute 4x for s and evaluate

Substitute

s= 4x

\textcolor{darkviolet}{A_\text{square}} = ({\color{#0000FF}{4x}})^2

PowProd

(a * b)^m=a^m* b^m

\textcolor{darkviolet}{A_\text{square}} = 4^2x^2

CalcPow

Calculate power

\textcolor{darkviolet}{A_\text{square}}=\textcolor{darkviolet}{16x^2}

Finally, let's find the ratio.

\boxed{\text{Ratio}}=\dfrac{\textcolor{darkorange}{A_\text{circle}}}{\textcolor{darkviolet}{A_\text{square}}}

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Substitute values and evaluate

SubstituteII

\textcolor{darkorange}{A_\text{circle}}={\color{#0000FF}{\textcolor{darkorange}{4\pi x^2}}}, \textcolor{darkviolet}{A_\text{square}}={\color{#009600}{\textcolor{darkviolet}{16x^2}}}

Ratio=4π x^2/16x^2

ReduceFrac

a/b=.a /4x^2./.b /4x^2.

Ratio=π/4

This tells us that the ratio is π4.

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