express the area A as a function of the width X

A Norman window has the shape of a rectangle surmounted by a semicircle. If the perimeter of the window is 22 ft, express the area *A* of the window as a function of the width *x* of the window.

I will label 2 sides as y and base as x

**π is pi**

Perimeter = **2y + x + ((2****π****r)/2)**

since there isn't really a base (x) at the top of the window I will remove x from the equation above.

I will then plug in **Radius as (x/2)**

p= **2y + ((2π(x/2))/2)**

= 2y + (πx)/2

22 = 2y + (πx)/2

Solving for y in above equation I came up with:

y = 11 - [(πx)/4]

Now for the Area, A(x)

A(x) = xy + (πr^2)/2

Plugging in Y and Radius obtained before my answer to this problem came to:

A(x) = x[11 - ((πx)/4)] + [(π (x/2)^2)/2]

any thoughts?

Re: express the area A as a function of the width X

Quote:

Originally Posted by

**NeoSonata** A Norman window has the shape of a rectangle surmounted by a semicircle. If the perimeter of the window is 22 ft, express the area *A* of the window as a function of the width *x* of the window.

I will label 2 sides as y and base as x

**π is pi**

Perimeter = **2y + x + ((2****π****r)/2)**

since there isn't really a base (x) at the top of the window I will remove x from the equation above.

I will then plug in **Radius as (x/2)**

p= **2y + ((2π(x/2))/2)**

= 2y + (πx)/2

22 = 2y + (πx)/2

Solving for y in above equation I came up with:

y = 11 - [(πx)/4]

Now for the Area, A(x)

A(x) = xy + (πr^2)/2

Plugging in Y and Radius obtained before my answer to this problem came to:

A(x) = x[11 - ((πx)/4)] + [(π (x/2)^2)/2]

any thoughts?

All your calculations are OK.

Now expand the bracket and collect like terms:

$\displaystyle A(x)= 11x - \frac \pi4 x^2 + \frac \pi8 x^2=11x-\frac \pi8 x^2$

... that's all!