Find the dimensions of a rectangle with maximum area that can be inscribed in a circle of a radius of 10.

Okay, so I know that I am going to need the Pythagorean theorem, where x^2+y^2=20^2 (20 is from the doubling of the radius which actually makes the hypotenuse of the rectangle.)

Up to there, I have no clue. I just know it has something involved with derivatives.

I would stick with x^2+y^2 =10^2 = 100

then the area = 2x*2y = 4 x y

A = 4 x y where y = (100-x^2)^.5

A = 4 x (100-x^2)^.5
dA/dx = 0 = 4 [-2x^2 (.5)(100-x^2)^-.5 +(100-x^2)^.5 ]

x^2 = 100 - x^2

2 x^2 = 100
x^2 = 50
x = about 7
y = about 7
area = about 4*49 = 196
You can do this all more carefully and accurately

To find the dimensions of the rectangle with maximum area that can be inscribed in a circle of radius 10, we can proceed as follows:

1. Start by representing the rectangle's length and width as variables. Let's call the length of the rectangle x and the width y.

2. Since the rectangle is inscribed in a circle, the length and width must form the hypotenuse of a right triangle inside the circle.

3. Use the Pythagorean theorem to relate the length, width, and radius of the circle. From your suggestion, we have x^2 + y^2 = 20^2.

4. Since we want to maximize the area of the rectangle, we need to express the area in terms of a single variable. The formula for the area of a rectangle is A = length * width = x * y.

5. Solve the Pythagorean equation in step 3 for one of the variables. Let's solve for y: y = sqrt(20^2 - x^2).

6. Substitute the expression for y from step 5 into the area equation from step 4. We get A = x * sqrt(20^2 - x^2).

7. The goal is to find the value of x that maximizes the area of the rectangle. To do this, we can take the derivative of A with respect to x and set it equal to zero. The critical points will correspond to potential maximums.

8. Differentiate A = x * sqrt(20^2 - x^2) using the product rule and chain rule. Then set the derivative equal to zero and solve for x.

9. Once you have the values of x that satisfy the derivative equaling zero, plug them into the equation for y from step 5 to find the corresponding values of y.

10. Check the endpoints of the interval by substituting x = 0 and x = 20 into the area equation to ensure there are no larger values.

11. Compare the areas of the rectangles calculated at the critical points and endpoints to determine the maximum area and the corresponding dimensions of the rectangle.

Remember, the derivative is used to find the maximum by finding the values of x where the slope of the function is zero. This is a basic overview of the process; for step-by-step calculations, you may consult a calculus resource or use a graphing calculator to aid you in solving the equation numerically.