A wire is 50 inches long. Where should the wire be cut so that it forms 2 pieces one made into a circle and one made into a square such that the area of the circle plus the area of the square in exactly 80 square inches?
"A wire is 50 inches long. Where should the wire be cut so that it forms 2 pieces one made into a circle and one made into a square such that the area of the circle plus the area of the square in exactly 80 square inches?"
Hmm!
I think your total area should be 800 square inches NOT 80 square inches.
Let x = side length of the square.
50 - x =Circumference of the circle.
[50 - x] / 2pi =Radius of the circle
x^2 + [(50 -x)/(2pi)]^2*pi = 800, solve for x
x =27.567 inches - one side of the square.
50 - 27.567 =22.433 inches - Circumference of the circle.
So that:22.433/2pi =3.57 inches - radius of the circle.
27.567^2 + [3.57^2*pi] =~800 sq.inches.
So, the 50-inch wire should be cut into 2 pieces:
27.567 inches - being one side of the square.
22.433 inches - being the circumference of the circle.
1. If x is the side length of the square then the circumference of the circle is 50 - 4x rather than 50 - x.
2. However, note that the maximum possible area that can be achieved by a piece of wire 50 inches long is when the whole wire is used to make a circle. This results in an area of \(\pi (\frac{50}{2\pi})^2 = 198.94... in^2\) i.e. much less than 800 in2.