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?

To find the dimensions of the rectangle with maximum area, we need to understand the relationship between the rectangle and the semicircle.

Let's start by visualizing the problem. Draw a semicircle with a radius of 5 cm and mark its center as point O. Now, draw a rectangle that has its base on the diameter of the semicircle and two vertices on the semicircle. Let the length of the rectangle be L and the width be W.

Since the rectangle has its base on the diameter of the semicircle, its width W is equal to the radius of the semicircle, which is 5 cm.

The length of the rectangle, L, can be found by subtracting 2W from the total diameter of the semicircle. The total diameter is equal to twice the radius, so it is 2 * 5 cm = 10 cm. Subtracting 2W gives us L = 10 cm - 2 * 5 cm = 0 cm.

Wait, this doesn't seem right. Our formula suggests that the length of the rectangle is 0 cm. However, this doesn't make sense because a rectangle with zero length wouldn't exist.

Let's think about it intuitively. When the rectangle is a square, its diagonal will be the same as the diameter of the semicircle. This is because the diagonal of a square divides it into two congruent right triangles, and the hypotenuse of each triangle is the diameter of the semicircle.

In our case, we have a rectangle that is not a square. If we decrease the length of the rectangle, the diagonal also decreases, which means it becomes smaller than the diameter of the semicircle. As a result, the rectangle no longer fits within the semicircle.

Therefore, the rectangle with maximum area must be a square, where the length L is equal to the width W. Since W is given as 5 cm, the dimensions of the rectangle with maximum area are 5 cm by 5 cm.

Try the same methodology as in the previous post.

http://www.jiskha.com/display.cgi?id=1288919599

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