Study Group Needs Your Help w/ Optimization Calculus Problem?

Question

Sorry to ask such a question, but our study group is at a loss as how to continue and our homework is due tomorrow. So here goes:

In order to receive credit we MUST use calculus techniques:

We have a piece of wire that is 100cm long and we're going to cut it into two pieces. One piece will be bent into a square and the other will be bent into a circle. Determine where the wire should be cut so that the enclosed areas will be at maximum. Note that it is possible to have the whole piece of wire go either to the square or to the circle

Answer 1

Let x be length of the wire used to make the circle. Then the amount left over is 100 - x, which will be used to make the square.

x is the same length as the perimeter of the circle. Find the radius using the formula for the circumference. Use the radius to get a formula for the area.

100-x is the perimeter of the square. What length will the side have then? Use this side length to find the area of the square.

Add the two formulas for the two shapes together to get a formula for the total area. This will be the function that you want to maximize. Don't forget the limits on your x. 0 ≤ x ≤ 100cm.

The answer will be x = 100. If you don't get that you did something wrong.

Good luck!

Answer 2

Let a be the length of wire to make the square.
Let b the remaining length, which is used for the circle.

a + b = 100

Let S be the area of the square, and C be the area of the circle.

S = (a/4)^2

Find the radius of the circle
b = 2*pi*r
r = b/(2 *pi)

C = pi * b^2/[(2*pi)^2] = b^2/(4*pi)

Rewriting it in terms of a,
C = (100 - a)^2/(4pi)

The total area (T) is S + C

T = (a/4)^2 + (100 - a)^2/(4pi)

The question now becomes, find a to maximize T.

You should be able to figure this one out now.

Answer 3

It's actually quite simple...

basically you have a 100cm long wire... then cut it to form a square in one and a circle in the other...

therefore we can say that

100cm=(circumference) + (total length of sides)
100cm=2pi*R+4s

that's formula no. 1

however, the problem is that we need to maximize the total area, so we get the total area A.

A=(area of circle)+(area of the square)
A=pi*R^2+s^2

getting the value of s in the first equation then substituting it to the latter formula, we get

A=pi*R^2+625-25pi*R+0.25pi^2R^2

taking the first derivative to get the maximum R, we have

A'=2pi*R-25pi+2(0.25)(pi^2)R=0

solving for R w have

R=7.001cm

to get s, we simply use the first equation

100=2pi*(7)+4s
s=14.0044cm

but the question is where the wire is to be cut... then, we simply get

4s=4(14.0044)=56.0177cm

so we cut it in the 56.0177cm mark where 56.0177 cm will go to make the square and 43.9823cm will be made a circle

Hope it helps..