Question

A Norman window has the shape of a rectangle surmounted by a semicircle. If the perimeter of the window is 24 ft, express the area A of the window as a function of the width x of the window.

Answer 1

This norman window looks like a rectangle with a semi circle on top of it.

Answer 2

the width "x" is the same as that of the diameter of the semi circle.

Answer 3

What?#4

Answer 4

Let me attach an image.

Answer 5

(πx/2)+3x =24
=> x=5.25
=> area = (5.25^2)+[(π/8)*(5.25^2)] =38.39
In general the equation is \[(x ^{2})[( 8+\pi)/8]\]

Answer 6

this solution is only correct if the semicircle is mounted by a square

Answer 7

Yes

Answer 8

Okay the perimeter of the window has two parts, the rectangle part and the half circle part. The half circle has a perimeter equal to half it's circumference which is \[\frac{1}{2}\pi x\]The rectangle part has a perimeter equal to x+2y where y is the height of the rectangle and x is the width. So our first equation is:\[\frac{1}{2}\pi x +x + 2y =24\]Now for the area. Again we have two parts, the half-circle and the rectangle part. The expression is:\[A=\frac{1}{2} \pi (x/2)^2+xy=\frac{\pi x^2}{8} + xy\]Now we want to express this in terms of x alone. We solve for y in terms of x from the first equation and get: \[y=12-(x/2)-\frac{1}{4}\pi x\]Inputting this into our area function we get:\[A(x)=\frac{\pi x^2}{8}+x(12-\frac{x}{2}-\frac{1}{4}\pi x)\]I'll leave it to you to simplify it.

Answer 9

Let l be the length of the rectangle
\[x+2l+(\pi x /3) = 24\]
=> \[l = 12-[ (3+\pi)/6)x]\]
=> Area = [(π/8)x^2]+{x*(12−[(3+π)/6)x])}

Answer 10

@naga: Your expression for perimeter is incorrect. Where do you get \[\frac{\pi x}{3}\] for the perimeter of the half-circle?

Answer 11

i need help with this question two :O