Maximizing Area of a Racetrack: What are the Dimensions for Optimal Performance?

[falcarius]

Homework Statement

The Question is:
"A 400km racetrack is to be built with two straight sides and semicricles at the ends. Find the dimensions of the track that encloses the maximum area."

The two long sides of the rectangle are written with >/= to 100m (each)
The straight side of the 2 semi circles is written with >/= to 20m (each)
(>/= means greater than or equal to, just in case)

Homework Equations

Area of semicircle = 1/2 * pr2
Area of rectangle = lw

The Attempt at a Solution

Well i tried but its always the beginning setting up of optimization problems that is the killer, the rest is always easy. It is a study Q for an upcoming test.

 

[danago]

Find a way to write the length of the straight section in terms of the radius of the end circles. To do this, you need to use the fact that the total perimeter is 400km.

 

[HallsofIvy]

In the semicircle ends, r= w/2 where w is the "width" of the rectangle. Since there are two ends, you really have one circle. The area you want to maximize is $\pi r^2= \pi/2 w^2$. As danago said, the circumference of the figure is 400 m so $2l+ \pi/4 w^2= 400$.