# Thread: Maximizing Area & Differentials

1. ## Maximizing Area & Differentials

A window consists of an open rectangle topped by a semicircle and is to have a perimeter of 288 inches. Find the radius of the semicircle that will maximize the area of the window.

I don't even know where to start, what does the picture even look like?

AND

The measurement of the edge of a cube is found to be 12 inches, with a possible error of 0.03 inch(a little bit of change in length). Use differentials to approximate the maximum possible propogated error in computing.
(a) the volume of the cube
(b) the surface area of the cube

2. Originally Posted by Th3On3Fr33man
A window consists of an open rectangle topped by a semicircle and is to have a perimeter of 288 inches. Find the radius of the semicircle that will maximize the area of the window.

I don't even know where to start, what does the picture even look like?
...
Hello,

I've attached a sketch of the window.

The area of this window consists of a rectangle and a semicircle:

$A=2r \cdot h + \frac12 \cdot \pi r^2$

The perimeter of the window is calculated by:

$p = 2r + 2h + \pi r$ which is a constant of 288''. Thus:

$h = 144 - r - \frac12 \pi r$

Substitute the variable h in the first equation by this term. You'll get a function A(r). Derivate this function and solve the equation A'(r) = 0 for r.

I've got $r = \frac{288}{4+\pi} \approx 40.327\text{''}$