Question

a rectangle is inscribed with its base on the x-axis and its upper corners on the semicircle defined by in quadrants i and ii. what is the height of the rectangle with greatest possible area?

Answer 1

The height of the rectangle with the greatest possible area is equal to 2 times the radius of the semicircle.

Step-by-step explanation:

To find the height of the rectangle with the greatest possible area, we can use the fact that the rectangle is inscribed with its base on the x-axis and its upper corners on the semicircle. Since the base is on the x-axis, the width of the rectangle is equal to the length of the x-axis segment between the two upper corners. Let's denote this length as 2x.

The height of the rectangle is then the vertical distance between the x-axis and the semicircle. Since the upper corners lie on the semicircle, we can find the height by subtracting the y-coordinate of the semicircle at the x-axis from the radius of the semicircle. Let's denote the radius as r. The y-coordinate of the semicircle at the x-axis is equal to -r, so the height of the rectangle is r - (-r) = 2r.

Therefore, the height of the rectangle with the greatest possible area is equal to 2 times the radius of the semicircle.