Answer to Question #132925 in Calculus for x

Question

A Norman window with a perimeter of 20 units has the shape of a rectangle surmounted by a
semi-circle. If its base is x units long and the height of the rectangle is y units long, write a mathematical model for its area in terms of x.

Answer 1

let P perimeter and S area

r = x/2 semi-circle where r radius

P = x+2*y +$\pi$ *x/2

y = (P -$\pi$ *x/2-x)/2 = (10 - $\pi$ *x/4- x/2)

S = x*y +$\pi$ *(x/2)²/2

S = x* (10 - $\pi$ *x/4- x/2)/2 + $\pi$ *(x/2)²/2 = 10x - $\pi$ *x²/8 -x²/4 + $\pi$ *x²/8 = 10x - x²/4