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The diagram shows a track composed of a rectangle with a semicircle on each end. The area of the rectangle is 2 comma 800 square meters. What is the perimeter of the​ track? Use 3.14 for pi.

70 m
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Part 1
The perimeter of the track is
  
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​(Round to the nearest tenth as​ needed.)

1 answer

To find the perimeter of the track, which consists of a rectangle and two semicircles at each end, we will follow these steps:

  1. Identify the dimensions of the rectangle:

    • Let's denote the length of the rectangle as \( L \) and the width (diameter of the semicircles) as \( W \).
    • The area of the rectangle is given by: \[ A = L \times W = 2800 \text{ m}^2 \]

  2. Calculate the perimeter of the track:

    • The perimeter consists of the lengths of the two long sides of the rectangle and the circumference of the two semicircles.
    • The circumference of a full circle is \( C = \pi D \), where \( D \) is the diameter. Since we have two semicircles, the total contribution from the semicircles will be the same as the circumference of a full circle with diameter \( W \): \[ \text{Circumference of two semicircles} = \pi W \]
    • Thus, the total perimeter \( P \) is given by: \[ P = 2L + \pi W \]

  3. Determine specific values:

    • Since no specific dimensions are provided for \( W \) or \( L \), let’s first express one variable in terms of the other from the area equation. If we rearrange: \[ L = \frac{2800}{W} \]
    • Substitute \( L \) into the perimeter equation: \[ P = 2 \left( \frac{2800}{W} \right) + \pi W \]
    • Using \( \pi \approx 3.14 \): \[ P = \frac{5600}{W} + 3.14 W \]

  4. Finding dimensions to calculate the perimeter:

    • If you need a specific perimeter, you would need specific dimensions \( L \) and \( W \). However, if we assume likely values based on common rectangular dimensions providing 2800 m² area, such as \( W = 40 , \text{m} \):
      • Then \( L = \frac{2800}{40} = 70 , \text{m} \)
      • Substitute into the perimeter formula: \[ P = 2(70) + 3.14(40) \] \[ P = 140 + 125.6 = 265.6 \]

Thus, the perimeter of the track is approximately 265.6 meters.

If you have specific dimensions or a different width/length ratio in mind, please provide them for a more accurate calculation.