Question

A wire 10 cm long is cut into two pieces, one of length x and the other of length
10 − x,
as shown in the figure. Each piece is bent into the shape of a square.

(a) Find a function that models the total area enclosed by the two squares. (Let x be the length of wire in cm that is bent into a square and A(x) be the area.)

(b) Find the value of x that minimizes the total area of the two squares.

I know (a) is A(x)= (x/4)^2 + ((10-x)/4)^2

I am only having trouble with part (b) please help!!

Answer 1

Set the derivative of \[\left(\frac{x}{4}\right)^2+\left(\frac{10-x}{4}\right)^2 \]to zero \[\frac{1}{8} (-10+x)+\frac{x}{8}=0\]and solve for x.
x = 5
The answer is two squares with perimeters of length 5cm each. The total area of the two squares is \[2*(1.25)^2=3.125 \text{ cm}^2 \]