A rectangle is inscribed in a circle of radius 10 inches. This means that all four corners of the rectangle are on the circle. Find the dimensions of the largest possible area of the rectangle. What...

A rectangle is inscribed in a circle of radius 10 inches. This means that all four corners of the rectangle are on the circle. Find the dimensions of the largest possible area of the rectangle. What is that area?

1. Draw a picture (sometimes a 2D and 3D picture is helpful)

2. Create relevant variables and label both pictures. Hint: What values do you have control over?

3. Write a function that represents the quantity you want to maximize. (This is called an objective function.) Give is a sensible name and use the variables you defined above.

4. If you have more than one variable, find relations that let you reduce down to just one variable. (These relations between variables are called constraints.)

5. Identify a domain for your one remaining variable. Try to make it a closed interval!! Justify this in physical terms. (Making a closed interval allows us to use the Extreme Value Theorem.)

6. Use Calculus to find the optimal value of this one-variable volume function