# AP Calculus-Optimization

• Dec 2nd 2007, 07:18 PM
Th3On3Fr33man
AP Calculus-Optimization
Hey I need to get these as fast as possible:

1. A rectangle has its base on the x-axis and its upper two vertices on the parabola y=12-x^2. What is the largest area the rectangle can have, and what are its dimensions?

2. You are planning to make an open rectangular box from an 8- by 15-in. piece of cardboard by cutting congruent squares from the corners and folding up the sides. What are the dimensions of the box of largest volume you can make this way, and what is its volume?

I can usually work with these once I have the correct equations (by finding the derivative and setting it = to 0) but I have a real problem with getting the actual equation...

(1) you want to maximize $\displaystyle A = xy = x(12-x^2)$
(2) Let $\displaystyle x = \text{side of square cut from corners}$. We want to maximize the volume $\displaystyle V$.
So $\displaystyle V = x(15-2x)(8-2x)$. Find critical points. I get $\displaystyle x = 6, \ x = \frac{5}{3}$. The maximum volume occurs at $\displaystyle x = \frac{5}{3}$.
Thus $\displaystyle V = \frac{2450}{27}$ and the dimensions are $\displaystyle \frac{35}{3}, \frac{14}{3}, \ \frac{5}{3}$.