Optimization semicircle problem

[jimen113]

[SOLVED] Optimization problem

Homework Statement

A Norman window has the shape of a semicircle atop a rectangle so that the diameter of the smicircle is equal to the width of the rectangle. What is the area of the largest possible Norman window with a perimeter of 28 feet?

Homework Equations

After solving for b and plugging b into the Area formula I cannot determine the local max. Is the algebra or derivative wrong?

The Attempt at a Solution

please see attachment

 

[jimen113]

Thanks for your suggestion, the perimeter of a circle is also also pi*diameter, so then a semi-circle P=1/2pi*d. Which is what I have for the 1st equation

 

[jimen113]

Homework Statement

A Norman window has the shape of a semicircle atop a rectangle so that the diameter of the smicircle is equal to the width of the rectangle. What is the area of the largest possible Norman window with a perimeter of 28 feet?

Homework Equations

After solving for b and plugging b into the Area formula I cannot determine the local max. Is the algebra or derivative wrong?

The Attempt at a Solution

please see attachment

Solution:
A(x)=14b-(4+pi/8)*b^2
A'(x)=14-(4+pi/4)*b
Solve for b: (56/4+pi)
Insert b into the original area formula A(x) and the area of the largest possible window=54.8897ft
The problem was set up correctly except that I rounded off to only one significant figure and that's why my original answer didn't match the answer provided in the book.