A Calculus student bought 20 m of flexible garden edging (shown in green). He plans to put two gardens in the back corners of his parents' property: one square and one in the shape of a quarter circle. He will use the edging on the interior edges (shown in green on the diagram).

Question

a) If x represents the section of edging used for the quarter circle, show that the total area for the two gardens can be modeled by the function below:
A(x)=x^2/π+(10−0.5x)^2.

*Image of a giant rectangular box with a green square in the top left corner and a green quarter circle in the top right*

b) How should the wise Calculus student split the edging into two pieces in order to maximize the total area of the two gardens? Remember, it could be entirely one of the two shapes.

Answer 1

apparently the side of the square is 10-.5x
(a) the 1/4 circumference of the circle is x, so the radius r can be found by
2πr/4 = x
r = 2x/π
So, the area of the 1/4 circle is 1/4 πr^2 = π/4 (2x/π)^2 = x^2/π

(b) to maximize the area, set its derivative to zero:
A(x)=x^2/π+(10−0.5x)^2 = (1/π + 1/4)x^2 - x + 100
This is a parabola which opens up, so it has no maximum, except at one of the ends of the interval. That means either the square or the circle has area zero.
???