The area of a circle is square inches. A sector of the circle has area square inches. Which choice is the length of the arc that bounds the sector?

Solution available with step-by-step explanation

Super-Hard SAT Math Question 44 | Free SAT Prep | Aniko

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Question 44·200 Super-Hard SAT Math Questions·Geometry and Trigonometry

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The area of a circle is $144\pi$ square inches. A sector of the circle has area $40\pi$ square inches.

Which choice is the length of the arc that bounds the sector?

When a sector area is compared with the whole circle, convert that comparison into a fraction first. Then apply the same fraction to the full circumference. This is often cleaner than trying to work from an angle that was never given directly.

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Find the radius from the full area

Since $\pi r^2=144\pi$ , the radius can be found before you think about the sector.

Use the same fraction twice

The sector is $\frac{40\pi}{144\pi}$ of the full circle, so its arc length is the same fraction of the full circumference.

Desmos Guide

Compute the radius and circumference

In Desmos, enter r = sqrt(144) and then C = 2*pi*r. This gives the full circumference of the circle.

Use the sector fraction

Enter (40/144)*C. The result matches the correct answer choice for the arc length.

Step-by-step Explanation

Find the circumference

From the circle area,

\pi r^2=144\pi.

So $r^2=144$ , which means $r=12$ . Therefore, the circumference is

2\pi r = 24\pi.

Find the sector fraction

The sector area is

\frac{40\pi}{144\pi}=\frac{5}{18}

of the full circle.

Apply that fraction to the circumference

The arc length is the same fraction of the full circumference:

\frac{5}{18}\cdot 24\pi=\frac{20\pi}{3}.

So the correct choice is $\frac{20\pi}{3}$ .

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Desmos Guide

Step-by-step Explanation

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Step-by-step Explanation