Question

Four feet of wire is to be used to form a square and a circle. How much of the wire should be used for the square and how much for the circle to enclose the maximum TOTAL area? (Hint: write an equation for the area and the perimeter to get an area equation with 1 variable).

Answer 1

put x = amount for square then 4 - x = amount for circle

Answer 2

then each side of the square will have length
\[\frac{x}{4}\] with area
\[(\frac{x}{4})^2\] and the circle with have circumference of 4 - x and area
\[\frac{(4-x)^2}{4\pi}\]

Answer 3

add them up to get a formula for the total area. take the derivative, set = 0 and solve to find the max