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11 days ago

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USING STRUCTURE An athletic facility is building an indoor track. The track is composed of a rectangle and two semicircles, as s

hown. A drawing shows an indoor track. It is shaped like a rectangle with a semi-circle on each of the shorter sides. The length of the rectangle is labeled x. The radius of each semi-circle is labeled r. A. Write a formula for the perimeter of the indoor track. $P=$ b. Solve the formula for $x$ . $x=$ Question 2 c. The perimeter of the track is 660 feet, and $r$ is 50 feet. Find $x$ . Round your answer to the nearest foot. $x=$ feet

1 answer:

AnnZ [9.1K]11 days ago

4 0

Answer:

Step-by-step explanation:

Refer to the attached digram.

The track's perimeter consists of the rectangle's perimeter plus the perimeters of the two semicircles.

The perimeter of a rectangle can be expressed as 2(x+r) where:

x is the length

2r represents the rectangle's width, equivalent to the semicircle's diameter.

The perimeter of one semicircle is calculated as 2πr/2, simplifying to πr.

Thus, the total perimeter contributed by the two semicircles is 2πr.

Therefore, the full track perimeter is expressed as 2(x+2r) + 2πr

where r indicates the semicircle's radius.

Expanding this gives us

Perimeter of the track = 2x + 4r + 2πr

which can also be represented as Perimeter of the track = 2(x + 2r + πr)

b) Given P = 2(x + 2r + πr), we need to rearrange the equation to isolate x.

P = 2x + 4r + 2πr

By subtracting 4r and 2πr from both sides, we get P - 4r - 2πr = 2x

Next, we divide each side by 2.

(P - 4r - 2πr) / 2 = 2x

Thus, x = (P - 4r - 2πr) / 2

c) Given

P = 600ft

r = 50ft

x = (600 - 4(50) - 2π(50)) / 2

x = (600 - 200 - 100(3.14)) / 2

x = 400 - 314 / 2

x = 86 / 2

x = 43ft

Therefore, rounding to the nearest foot, the value of x is 43ft

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