A rectangle is inscribed in a semicircle with radius 8. The variable x is half the length of the rectangle. Write an expressions for the perimeter and area of the rectangle in terms of x.
Draw the diagram. It should be clear that the height of the rectangle is
√(8^2 - x^2)
Now you have the height and the length (2x), so the rest is a cinch.
To find the perimeter of a rectangle, you sum the lengths of all its sides. In this case, the rectangle's length is 2x, since x is half the length. The width of the rectangle is the same as the radius of the semicircle, which is 8. So, the perimeter (P) can be expressed as:
P = 2(2x) + 2(8)
P = 4x + 16
To find the area of a rectangle, you multiply its length by its width. Again, the length is 2x and the width is 8. So, the area (A) can be expressed as:
A = 2x * 8
A = 16x
So, the expressions for the perimeter (P) and area (A) of the rectangle in terms of x are:
P = 4x + 16
A = 16x