Question 1098711: A wire 10 meters long is to be cut into two pieces. One piece will be shaped as a square, and the other piece will be shaped as a circle. The figure shows 10m as the full length up and down. On the other side of the line it shows 4x, and 10-4x. It shows a square with an x, and a circle with nothing next to it.
(a) Express the total area A enclosed by the pieces of wire as a function of the length x of the side of the square.
(b) What is the domain A?
(c) Graph A=A(x). for what value of x is A the smallest?
My book doesn't show clearly how to set up this problem. I hope I was clear enough with the dimensions. I didn't know which category to post as it is in functions and their graphs, but didn't want to put it in the function category that said NOT graphing. Sorry if I have misplaced or not been clear. I will add on if needed. I don't just post because I want the answer, I truly want to know how to set the problem up so that I can successfully do the problem right on homework, and on a test. Thank you so much for taking the time to read my question and help show me how to do my homework problem! it's very appreciated!
Answer by Boreal(15235)   (Show Source):
You can put this solution on YOUR website! Using the notation there, the square has area x^2 and the circle's area is f(radius), and the radius can be found by taking 10-4x, the circumference, and dividing by 2pi, so the area of the circle is pi^2, after reducing the radius to (5-2x)/pi
The sum of these areas is x^2+[pi*(5-2x)^2/pi^2]
Take the derivative and set equal to 0: 2x+(1/pi)(2(5-2x)(-2))=0
This is 2x + (1/pi)(4x-20)=0
multiply by pi
2x*pi+4x-20=0=x*pi+2x-10
x(pi+2)=10
x=10/(pi+2)
Minimum is when A=3.5 m^2 and x=1.4 m
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