How to Maximize Area for a 1-km Racetrack with Semicircles?

[pacman99]

Hi,

I just needed help starting off this problem:

"A 1-km 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."

There was a similar question which I did before this which involved a Norman window. My idea was, in order to make everything in terms of one variable, let's say the width, I made radius equal to half of the width. The problem is that I DO get an answer but my textbook says that the answer is "a circular track with a radius of 1/2pi km". I find that weird because it says "racetrack is to be built with two straight sides..."

Anyways I just wanted to know, how should I set up the perimeter and area equations? I know the perimeter is 1km but should I solve for y by making r = half of y (width) or should i keep two variables and try to work with them? (width & radius)

Thanks!

 

[NSX]

hehe
fun fun

I just did this right now, again; I'm new to the LaTeX, so bare with my text (where there is space with no operators, this means multiply):

I called the radius of the circle r, and the straight part of the track x.

Circumference = C = 2 pi r ; r = radius
Area = A = pi r^2

Total Distance = d = 2 x + 2 pi r = 1 => x = 1/2 - pi r (1)

Area = A = pi r^2 + 2 r x (2)


Sub (1) into (2):

A = pi r^2 + 2 r (1/2 - pi r)

A = r - pi r^2

Take derivative of A w/r.t. r to find when the slope is 0 (since this means max or min):

dA/dr = 1 - 2 pi r

set dA/dt = 0:

1 - 2 pi r = 0 => r = 1 / (2 pi)

Now, you can do the first derivative test, and you see that for r < 1 / (2pi), the slope is positive, and for r > 1 / (2pi), the slope is negative.
Therefore maximum area at a track with a radius of 1 / (2 pi). Furthermore, you can find what the straight portion of the track should be by substituting r into equation (1).

Note that when you substitute r into equation (1), you get:

x = x = 1/2 - pi (1 / (2 pi)

The pis cancel and the 1/2 negate each other, meaning that the x, or straight part of the track should be zero to yield optimal results from the conditions given.

No x, means no straight part, and that means all circle

 

[M98Ranger]

Hi there,

To start off, let's define some variables to help us solve this problem:

- r: radius of the semicircles at the ends of the track
- w: width of the straight sides of the track
- l: length of the straight sides of the track

Now, let's think about what we know from the problem:

- The total perimeter of the track is 1 km, which means that the sum of all the sides must equal 1 km.
- The track has two semicircles at the ends, which means that the total length of the curved parts must be equal to the circumference of a circle with radius r, or πr.
- The straight sides have a length of l, which means that the total length of the straight parts must be equal to 2l.

Using this information, we can set up the following equations:

Perimeter: 2l + πr + πr = 1 km
Area: A = l * w + πr^2

Now, we need to optimize the area by finding the maximum value. In order to do this, we can use the first equation to solve for l in terms of r, and then substitute it into the second equation. This will give us an equation for the area in terms of only one variable, r.

Solving for l: 2l + 2πr = 1 km
l = (1 km - 2πr)/2

Substituting into the area equation: A = [(1 km - 2πr)/2] * w + πr^2

Now, we can take the derivative of this equation with respect to r and set it equal to 0 to find the critical points. From there, we can determine which value of r will give us the maximum area. Remember to also check the endpoints of the interval (0 < r < 1/2 km) to make sure you have the absolute maximum.

I hope this helps get you started on solving this optimization problem. Good luck!