Question

Use optimization techniques to answer the questions in Exercises 25–30.
Find the area of the largest rectangle that fits inside a circle of radius $10$.

Short answer

The area of the rectangle is $200$.

Step-by-step solution

Step 1. Given Information.

Radius of circle is $10$.

Step 2. Form an equation.

Let x be the length of the given rectangle.

From the given information,

Diameter$=10\times 2$

$=20$

Step 3. Use Pythagoras theorem to find x. 

Using Pythagoras theorem,

$$\begin{gathered} x^2+x^2={20}^2 \\ 2x^2=400 \\ {x}^2=200 \end{gathered}$$

Step 4. Find the area of rectangle.

The area of the rectangle which fits inside the area of the circle is the area of the square

$x^2=200$