Question

A circular pond is to be surrounded by a gravel path. Use the diagram to find the square feet of gravel needed if we know the pond has a radius of 200 ft and the radius of the whole space is 250 ft?

A. 86,270 ft2
B. 57,234 ft2
C. 70,686 ft2
D. 65,523 ft2

Answer 1

Let R = 250 ft. & r = 200 ft.

Now, Area of Gravel Path =
= πR² - πr²
= π(R²-r²)

Area of Gravel Path =
π ( 250² - 200² )
π ( 62500 - 40000 )
π ( 22500 )

Taking π as 3.14


`π * 22500 \\ 3.14 * 22500 \\ (314)/(100) * 22500 \\ 314 * 225 = 70650 \: f {t.}^(2)`
As Option is nearest to our answer,
that means it will be right.

So,
Answer will be (c) 70686 ft.²

Answer 2

C. 70,686 ft²

Step-by-step explanation:

There are a couple of ways to get there.

Fairly straightforward is subtracting the pond area from the overall area of path and pond. The area of a circle is given by ...

... A = πr²

so that approach looks like ...

... A = π·250² -π·200² = 196,349.5 - 127,663.7 ≈ 70,685.8 ≈ 70,686 . . . ft²

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You can factor π from the above expression to make the arithmetic a little easier:

... A = π(250² -200²) = π(62500 -40000) = 22500π ≈ 70,686 . . . ft²

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You can further factor the difference of squares to give:

... A = π(250² -200²) = π(250 -200)(250 +200) = π·50·450 ≈ 70,686 . . . ft²

You might notice that this latter formulation is the product of the length of the centerline of the path and its width.