if the perimeter of the window is 30ft, express the area A of the window as a function of the width x of the w

Question

A norman window has the shape of a rectangle surmounted by a semicircle. if the perimeter of the window is 30ft, express the area A of the window as a function of the width x of the window.

Answer 1

Window is rectangule with semicircle shape.
Say x= width of rectangle,
b = height of rectangle
So, x + 2b + pi(0.5x) = 30ft
x + 2b + 1.57x = 30
2.57x +2b = 30
b = 15 - 1.285x

Area, A = (x)(b) + 0.5pi(0.5x)^2
A = (x)(b) + 0.3925x^2
A = (x)(15-1.285x) + 0.3925x^2
A = 15x - 0.8925x^2

Answer 2

Let h be the height of the rectangle part of the window, as if it had no semicircle on top. The perimeter is then
2h + x + pi/2 x = 30, or
2 h + (1 + pi/2)*x = 30
Therefore, h = (30 - (1 + pi/2)x)/2.
The area is that of the rectangle and of the semicircle. The semicircle's area is half that of a circle with diameter x, or radius x/2, so it is
(pi (x/2)^2)/2, or
pi * x^2 / 8
and the rectangle has area
x * h = x*(30 - (1 + pi/2)x)/2
= 15 x - (1 + pi/2) x^2 / 2
= 15 x - (4 + 2 pi) x^2 / 8
so the window has area
A = 15 x - (4 + 2 pi) x^2 / 8 + pi * x^2 / 8
= 15 x - (4 + pi) x^2 / 8

Additionally, if you want to maximize the area of the window, differentiate it to get
dA/dx = 15 - (4 + pi) x / 4
Setting this to zero, we get
15 - (4 + pi) x / 4 = 0
15 = (4 + pi) x / 4
x = 60 / (4 + pi)
which is approximately 8.4014873

Answer 3

the area of the window is a functionality of the top as nicely because of the fact the width. the area of the rectangle is H x W the area of the semicircle is pi * W^2 / 8 upload those 2 together and you get the area of the window

Answer 4

Area of circle = (pi) (r^2).
r = (x/2)
Area of semicircle: [(pi) (x/2)^2] / 2
the remaining area will be (x) (long leg)

{[(pi) (x/2)^2] / 2}+ [(x) (long leg)] = 30

Answer 5

2x + 2w = 30 (perimeter)

xw = a (Area)