Question 84·Medium·Geometry and Trigonometry

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A circular fountain has radius $12$ feet. Each morning, a portion of the fountain is cleaned, forming a sector with area $72\pi$ square feet.

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

When a sector’s area is given, first find the fraction of the full circle it represents by dividing by $\pi r^2$ . Then apply that same fraction to the circumference $2\pi r$ to get the arc length.

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Compare areas

Compute the full circle’s area using $A=\pi r^2$ , then compare the sector’s area to that total.

Same fraction idea

The sector’s area is the same fraction of the circle’s area as its arc length is of the circle’s circumference.

Circumference

Once you know the fraction, multiply it by the circumference $2\pi r$ to get the arc length.

Desmos Guide

Compute the full circle area

Enter r = 12 and A = pi*r^2 to get the full area.

Find the sector fraction

Enter f = (72*pi)/A to find the fraction of the circle the sector represents.

Compute the arc length

Enter C = 2*pi*r and L = f*C.

Match L to the answer choices.

Step-by-step Explanation

Find what fraction of the circle is cleaned

The area of the full circle is $\pi r^2 = \pi(12)^2 = 144\pi$ .

So the cleaned sector is the fraction

\frac{72\pi}{144\pi} = \frac{1}{2}

of the circle.

Use the same fraction of the circumference

The circle’s circumference is $2\pi r = 2\pi(12)=24\pi$ .

The arc length bounding the cleaned sector is

\frac{1}{2}(24\pi)=12\pi.

Therefore, the arc length is $12\pi$ .

Question 84·Medium·Geometry and Trigonometry

Strategy

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Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation