A rectangle is constructed with its base on the diameter of a semicircle with radius 5 cm and with two vertices on the semicircle. What are the dimensions of the rectangle with maximum area?

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To find the dimensions of the rectangle with maximum area, we can proceed as follows:

Step 1: Understand the problem.
We have a semicircle with a radius of 5 cm. We need to construct a rectangle with its base on the diameter of the semicircle and two of its vertices on the semicircle. The goal is to find the dimensions (length and width) of the rectangle that maximize its area.

Step 2: Define the variables.
Let's denote the length of the rectangle as L and the width as W.

Step 3: Visualize the problem.
Draw a semicircle with a radius of 5 cm. Now, construct a rectangle with its base on the diameter of the semicircle. Two of its vertices lie on the semicircle. This rectangle will have a width equal to the radius of the semicircle (W = 5 cm) and a length equal to the diameter of the semicircle (L = 2 * 5 cm = 10 cm).

Step 4: Understand the formula for the area of a rectangle.
The area of a rectangle is given by A = L * W.

Step 5: Calculate the area of the rectangle.
Substitute the values of L and W into the area formula: A = 10 cm * 5 cm = 50 cm^2.

Therefore, the rectangle with maximum area has dimensions of 10 cm for the length and 5 cm for the width. Its area is 50 square centimeters.