Question

If a running track of 500 ft. is to be laid out enclosing a playground, the shape of which is a rectangle with a semicircle at each end, then the length of the rectangular portion such that the area of the rectangular portion is maximum is (in feet).

• A. \( 100 \)
• B. \( 125 \)
• C. \( 150 \)
• D. \( 200 \)

Hint:

For maximum area problems with constraints, express the function in terms of a single variable and use differentiation to find the critical points.

Answer 1

The Correct Option is B

We are asked to maximize the area of the rectangular portion of a playground enclosed by a running track with a total perimeter of 500 ft. Step 1: Identify Variables Let the length of the rectangular portion be \( L \) and the width be \( 2R \), where \( R \) is the radius of the semicircles. Since there are two semicircles at each end, their total circumference equals the circumference of a full circle, which is \( 2\pi R \). Step 2: Perimeter Equation From the given total perimeter condition, \[ L + 2R + L + 2\pi R = 500 \] Simplifying, \[ 2L + (2 + 2\pi)R = 500 \] \[ 2L + 2(1 + \pi)R = 500 \] Dividing the entire equation by 2, \[ L + (1 + \pi)R = 250 \] Step 3: Area of the Rectangular Portion The area of the rectangular portion is: \[ A = L \times 2R \] From the perimeter equation: \[ L = 250 - (1 + \pi)R \] Now, \[ A = 2R[250 - (1 + \pi)R] \] Expanding: \[ A = 500R - 2(1 + \pi)R^2 \] Step 4: Maximizing the Area To maximize the area, take the derivative and set it equal to zero: \[ \frac{dA}{dR} = 500 - 4(1 + \pi)R \] Set the derivative equal to zero: \[ 500 - 4(1 + \pi)R = 0 \] \[ R = \frac{500}{4(1 + \pi)} \] Since \( \pi \approx 3.14 \), \[ R = \frac{500}{4 \times 4.14} = \frac{500}{16.56} \approx 30.2 \] Step 5: Finding \( L \) Using \( L = 250 - (1 + \pi)R \), \[ L = 250 - 4.14 \times 30.2 \] \[ L \approx 250 - 125 \] \[ L \approx 125 \] Step 6: Final Answer  Correct Answer: (2) \ 125