A wire 10 meters ong is to be cut into two pieces. One piece wil be shaped as a square, and the other piece will be shaped as a circle. (piece one is 4x, piece two is 10-4x)

Question

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 of A?

c) Graph A=A(x). For what value of x is A smallest?

Answer 1

area square = x^2
diameter circle = d = (10-4x)/pi
so area circle = pi d^2/4
= (1/4pi)(10-4x)^2

A = x^2 +(1/4pi)(100 -80x +16 x^2)
= (1/pi)pi x^2 + (1/pi) (25-20x+4x^2)
= (1/pi)( (pi+4)x^2 -20 x + 25)
that is a parabola, look for zeros and vertex
note that negative A is meaningless as is negative x.
negative 10-4x is also meaningless so right off the bat you know x>10/4 is outside the domain
so x can only be between 0 and 10/4