A rectangle is constructed with its base on the diameter of a semicircle with radius 25 and with its two other vertices on the semicircle. What are the dimensions of the rectangle with maximum area?

Step 1 ：Let the length of the rectangle be x and the width be y. The base of the rectangle is on the diameter of the semicircle, so we can use the Pythagorean theorem to relate x and y to the radius of the semicircle (25): \(x^2 + y^2 = (2 * 25)^2\)

Step 2 ：We want to maximize the area of the rectangle, which is A = x * y. We can rewrite the area equation in terms of x or y using the Pythagorean theorem: \(A = -x\sqrt{2500 - x^2}\)

Step 3 ：Find the maximum area using calculus by taking the derivative of the area equation with respect to x: \(\frac{dA}{dx} = \frac{x^2}{\sqrt{2500 - x^2}} - \sqrt{2500 - x^2}\)

Step 4 ：Find the critical points by setting the derivative equal to zero: \(-25\sqrt{2}, 25\sqrt{2}\)

Step 5 ：Choose the critical point that corresponds to the maximum area: \(x = -25\sqrt{2}\)

Step 6 ：Find the corresponding value of y: \(y = -25\sqrt{2}\)

Step 7 ：\(\boxed{\text{Final Answer: The dimensions of the rectangle with maximum area are approximately } 25\sqrt{2} \text{ by } 25\sqrt{2}}\)