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 + ((2πr)/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?

Quote:

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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 + ((2πr)/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?

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All your calculations are OK.

Now expand the bracket and collect like terms:



... that's all!