Q. 21 - Calculus

Question

Find dimensions for each shape in Exercises 21–24 so that the total area enclosed is as large as possible, given that the total edge length is 120 inches. The rounded shapes are half-circles, and the triangles are equilateral.

Step-by-Step Solution

Answer

The dimension for the given shape is $r=\frac{60}{\mathrm{\pi }}$ where r is the radius of the circle.

Step 1. Given Information.

Total edge length$=120 inches$

The rounded shapes are half-circles, and the triangles are equilateral.

Step 2. Find the perimeter of semi circle.

Let r be the radius of the semi-circle.

There are two semi circles in the given figure.

So the perimeter of the circular part would be $2\mathrm{\pi r}$.

Step 3. Find the radius.

To find the total area enclosed is as large as possible, let the horizontal straight edge have length zero.

so,

$$\begin{gathered} 2\mathrm{\pi r}=120 \\ \mathrm{\pi r}=\frac{120}{2} \\ \mathrm{r}=\frac{60}{\mathrm{\pi }} \end{gathered}$$