Answer to Question #136452 in Calculus for Gabie

Question

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

Answer 1

The rough diagram of the question is given below

the perimeter of the window is given as 20 units.

Radius of the semicircle will be half of the length of AB.

i.e. radius = x/2 units

$Perimeter =AB + BC+ AD+ length \space of \space the \space arc \space CD\\or, x+y+y+ \frac{\pi x }{2}=20\\or, x+2y+\frac{\pi x }{2}=20\\or,2y=20-(x+\frac{\pi x }{2})\\or, y= \frac {20-(x+\frac{\pi x }{2})}{2}\\or, y= 10- \frac{x}{2}-\frac{\pi x }{4}$

Now Area of the window is given by

$A =$ area of rectangle ABCD + area of semicircle surmounted on CD

$or, A = xy + \frac{1}{2}*\pi*(\frac{x}{2})^2\\or, A = xy+\frac{\pi x^2}{8}\\or, A= x* (10- \frac{x}{2}-\frac{\pi x }{4}) + \frac{\pi x^2}{8}\\or, A = 10x-\frac{x^2}{2}-\frac{\pi x^2}{4}+\frac{\pi x^2}{8}\\or, A = 10x-\frac{x^2}{2}-\frac{\pi x^2}{8}$

Therefore mathemtical model of the area in terms of x is given by

$$A = 10x-\frac{x^2}{2}-\frac{\pi x^2}{8} \space \space \space \space (ANSWER)$$