A running circuit is to be constructed in the shape of a rectangle with two semicircles attached at the east and west ends of the path (see figure). A picnic area will be set in the center of the circuit. If we want the total length of the circuit to be 2000 m (2km ), what will be the dimensions (r and L) that make the picnic area maximum? Give you answer in meters.

Question

A running circuit is to be constructed in the shape of a rectangle with two semicircles attached at the east and west ends of the path (see figure). A picnic area will be set in the center of the circuit. If we want the total length of the circuit to be 2000 m (2km ), what will be the dimensions (r and L) that make the picnic area maximum? Give you answer in meters.

👇

2 answers
(click or scroll down)

👇

Answer 1

Steve answered
8 years ago

I will assume that we are just measuring the inner boundary of the path. Thus, the two lengths L and a circle of radius r has perimeter

p = 2L + 2πr
The inner picnic area is

a = L(2r) = (1000-πr)(2r)
= 2000r - 2πr^2

da/dr = 2000 - 4πr
da/dr=0 when r = 500/π

So, the picnic area is 500 by 1000/π

As usual, the length is divided equally between the lengths and the (curved) widths.

Answer 2

Explain Bot answered
11 months ago

To find the dimensions (r and L) that make the picnic area maximum, we can use calculus and optimization techniques.

Let's assume the length of the rectangle is L meters and the radius of the semicircles is r meters.

The total length of the circuit can be expressed as:

Length_of_Rectangle + 2 * Circumference_of_Semicircle = 2000 meters

The length of the rectangle is L meters, and the width of the rectangle is r meters (since a rectangle's length is the longer side).

Length_of_Rectangle = L meters, Width_of_Rectangle = r meters

The circumference of a semicircle with radius r can be expressed as:

Circumference_of_Semicircle = πr meters

So, the equation becomes:

L + 2πr = 2000 (Equation 1)

Now, let's express the area of the picnic area (rectangle area minus the semicircular areas):

Area_of_Picnic_Area = Length_of_Rectangle * Width_of_Rectangle - 2 * (Area_of_Semicircle)
= L * r - 2 * (πr^2/2)
= L * r - πr^2

To maximize the area, we can find the critical points of the function and check for the maximum. Taking the derivative of the area function with respect to r, we get:

d(Area_of_Picnic_Area)/dr = L - 2πr

The critical point occurs where the derivative is zero. So, setting the derivative equal to zero:

L - 2πr = 0
2πr = L
r = L/(2π) (Equation 2)

Substituting equation 2 back into equation 1:

L + 2π(L/(2π)) = 2000
L + L = 2000
2L = 2000
L = 1000 meters

Now that we have the length of the rectangle, we can find the radius of the semicircles by substituting back into equation 2:

r = 1000 / (2π)
r ≈ 159.154943 meters

Therefore, to make the picnic area maximum, the dimensions are approximately L = 1000 meters and r ≈ 159.154943 meters.