can u help me start this?

Question

A wire 5 ft long is to be cut into two pieces. One piece is to be bent into the shape of a circle and the other into the shape of a square. Where should the wire be cut so that the sum of the areas of the circle and square is
(a) a maximum
(b) a minimum

Answer 1

Let the radius of the circle be r.
Let the length of the side of the square be x.

Perimeter, P = 2(pi)(r) + 4x = 5
4x = 5 - 2pi(r)
x = [5 - 2pi(r)]/4

Area, A = (pi)(r^2) + x^2
A = pi(r^2) + {[5 - 2pi(r)]/4)^2
A = pi(r^2) + {25 - 20pi(r) + 4(pi^2)(r^2) / 16}
A = pi(r^2) + (25/16) - (20pi/16)(r) + (pi^2/4)(r^2)
A = [(pi + pi^2/4)(r^2) + (25/16) - (20pi/16)(r)

Max Area => dA/dr = 0
dA/dr = 2(pi + pi^2/4)r - (20pi/16)
2(pi + pi^2/4)r - (20pi/16) = 0
2(pi + pi^2/4)r = (20pi/16)
(pi + pi^2/4)r = (10pi/16)
(4pi + pi^2)r = (10pi/4)
r = (5pi/2) / (4pi + pi^2)

d2A/dr2 = 2(pi + pi^2/4) > 0 => minimum area

Answer 2

Start by letting the cut off piece to form the circle be x ft.

Then you have (5-x) to be bent into the square.

Get the length of the square side.


Get the radius of the circle.
Hint ( Circumference is x)

Then form an equation for the total area of the square and circle (based on the side of the square and raduius of circle)

Answer 3

use x for the square, 5-x for the circle.
so a side of the square is x/4
the radius of the circle is (5-x)/2pi
the areas are then x^2/16 and pi/4 (5-x)^2
sum differentiate set to 0. get 0 and 5 as extrema.
take 2nd derivative to see which is min, which max.

Answer 4

Since you only want me to start it for you, I will give you the formulas I would consider as important.

First, we know that we have only 5ft of string to use. This means the perimeters of the circle and the square must add to a total of 5 ft.

You will also need to think of how you can get the area of the circle and the area of the square to be related to the perimeters of both. Then you can relate area directly to the 5ft of string you have available.

Answer 5

It's a trick question. All you have is 5 feet of wire no matter where you cut it to bend it into shapes all you will ever have is 5 feet of circumference. The area will remain a constant.

Answer 6

NO