# Simple Calculus Word Problem (using derivatives to anaylze function models)

Hello, new here, first post. Just need some help with homework.

Question One

## Homework Statement

This norman window is made up of a semicircle and a rectangle. The total perimeter of the window is 16 cm. What is the maximum area?

**
* * <<< Semicircle
*****
| | <<< Rectangle
L | |
______
D

## Homework Equations

P (total) = 2L + D + (pi * d)

A (total) = D * L + (pi(d/2)^2)/2)

## The Attempt at a Solution

What I did was using this equation:
16 = 2L + D + ((pi * d)/2)
L = 8 - d/2 - ((pi * d)/4)

A = D (8 - d/2 - ((pi * d)/4)) + (pi (d/2)^2)/2)
A = 8d - (d^2)/2
A' = 8 - d
Let 0 = A' to find critical value
then 8 = d.

When I sub that back into the original equation, I get L as a value less than 8, which doesn't make sense. (I think it works out to be L = 4 - pi)

I'm pretty much lost, sorry if this is too messy to read, any help would be appreciated. Thanks

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Dick
Homework Helper
You made a great start. But where did you get "16 = 2L + D + ((pi * d)/2)"?? The "/2" wasn't in your original expression for P. You are just making algebraic mistakes.

I think you're doing fine up until this point:

A = D (8 - d/2 - ((pi * d)/4)) + (pi (d/2)^2)/2)

Which should simplify into

$$A = 8*D - \frac{D^2}{2} - \frac{pi*D^2}{4} + \frac{pi*D^2}{8}$$

You would then go on to take the derivate and then set it to zero and solve for your D value

I've been beaten =(

Last edited:
You made a great start. But where did you get "16 = 2L + D + ((pi * d)/2)"?? The "/2" wasn't in your original expression for P. You are just making algebraic mistakes.
ah sorry, its actually supposed to be "/2", that way its half the area, sorry the drawing didnt show up. its supposed to be a semi-circle connected to a rectangle.

I think you're doing fine up until this point:

A = D (8 - d/2 - ((pi * d)/4)) + (pi (d/2)^2)/2)

Which should simplify into

$$A = 8*D - \frac{D^2}{2} - \frac{pi*D^2}{4} + \frac{pi*D^2}{8}$$

You would then go on to take the derivate and then set it to zero and solve for your D value

I've been beaten =(
sorry, that was a typing error as well haha.

A = L * D + (pi*d)/2
which becomes

A = 8 - d/2 - ((pi * d)/4)

this still doesnt work...I think i'm using the wrong equations somehow

Dick