A circular fountain with a diameter of 8 feet is positioned in the center of a square lawn… — SAT

A circular fountain with a diameter of $8$ feet is positioned in the center of a square lawn. The lawn has a side length of $20$ feet. A concrete walkway of width $2$ feet is built directly around the fountain. What is the area, in square feet, of the remaining grass region of the lawn?

$400 - 36\pi$ Answer
B
$400 - 16\pi$
C
$400 - 24\pi$
D
$400 - 100\pi$

Answer

400 - 36\pi

The area of the square lawn is

20^2 = 400

square feet. The fountain has a radius of

4

feet (half of the

8

-foot diameter), and the walkway adds another

2

feet to the radius, making the total radius of the combined circular area

6

feet. The area of this circular region is

\pi \times 6^2 = 36\pi

square feet. Subtracting this from the total area of the square lawn gives the remaining grass area of

400 - 36\pi

square feet.

Step-by-Step Solution

Calculate the total area of the square lawn.

Area of the lawn is

20 \times 20 = 400

square feet.

The area of a square is calculated by squaring its side length.

Find the combined radius of the fountain and the walkway.

The radius of the fountain is

8 \div 2 = 4

feet. Adding the

2

-foot width of the walkway gives a combined radius of

4 + 2 = 6

feet.

The radius is half of the diameter. The walkway surrounds the fountain, so its width must be added to the fountain's radius to find the outer boundary's radius.

Calculate the combined area of the fountain and the walkway.

Area of the combined circular region is

\pi \times 6^2 = 36\pi

square feet.

The area of a circle is given by

\pi r^2

, where

r

is the radius.

Subtract the combined circular area from the total area of the lawn to find the remaining grass area.

The remaining area is

400 - 36\pi

square feet.

The remaining grass area is the total square area minus the area of the inner circular region that contains the fountain and the walkway.

Key Concept

Area of composite shapes involving squares and circles, and scaling properties.

Estimated Time:1m 30s

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