Next, we can simplify and square both sides: x^2 = 4(1 − x^2 ). Now, we can solve this easily. We
get x = ± . Next, we find the y-coordinate: y = ± . There are thus four possible answers
but, if we look at the picture, the answer is obviously the point . We could verify
that this is a minimum by taking the second derivative, but that will be messy. It is simpler to
use the calculator to check the sign of the derivative at a point on either side of the answer, or
to graph the equation for the distance.
- r = inches
First, let’s draw a picture.
Call the width of the window 2r. Notice that this is the diameter of the semicircle. Call the
height of the window h. The area of the rectangular portion of the window is 2rh, and the
perimeter is 2r + 2h. The area of the semicircular portion of the window is , and the
perimeter is = πr. Therefore, the area of the window is A = 2rh + , and the perimeter
