SOLUTION: The question in the book reads "A piece of wire 20 inches long is to be cut into two pieces, one of which will be bent into a circle and the other into a square. How long should ea
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-> SOLUTION: The question in the book reads "A piece of wire 20 inches long is to be cut into two pieces, one of which will be bent into a circle and the other into a square. How long should ea
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Question 106776This question is from textbook College Algebra
: The question in the book reads "A piece of wire 20 inches long is to be cut into two pieces, one of which will be bent into a circle and the other into a square. How long should each piece be to minimize the sum of the areas?". So one section of the wire could be represented as "x" and the other piece as "20-x". That's all I can come up with. Any help would be greatly appreciated!! This question is from textbook College Algebra
You can put this solution on YOUR website! The total length is 20 inches.
Let the two lengths be x and 20-x inches
Using calculus:
The area of the square is (x/4)^2 and the area of the circle is pi*radius^2.
Now the circumference = 2*pi*r and C=20-x.
Hence r=(20-x)/(2*pi)= 10/pi-x/2pi
So total area is A=(x/4)^2+pi*(10/pi-x/2pi)^2=(x^2)/16+pi*(100/(pi)^2+x^2/4pi^2-10x/pi^2)
A= x^2/16 + 100/pi + x^2/4pi - 10x/pi
Maximum or minimum values are given when dA/dx=0 and so:
2x/16+2x/4pi - 10/pi = 0
x/8+x/2pi - 10/pi = 0
Multiply by 8pi
x*pi + 4x - 80 = 0 = 0
x*(pi + 4) - 80 = 0
x = 80/(pi + 4)
x = 80/(22/7 + 28/7) = 80/(50/7) = (8*7/5 = 56/5 = 11.2
So the lengths are 11.2 inches and 8.8 inches ANS
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