Pre Calculus question

Question

1. a sheet of metal 20 by 16 inches is to be used to make a open box. squares of equal sides x are cut out of each corner then the sides are folded to make the box. find the value of x that makes the volume maximum

2. A farmer has 150 feet of fencing he is using to create a lot for his pack of llamas. he wants to keep them secluded in this fence which boarders his barn so they can no longer spit on his wife’s flowers. what is the maximum area of the lot he can make for them? what are the dimensions of the lot?

3. Write the equation of a polynomial with roots A and B such that a+b=7 and 1/a + 1/b=5 what are the roots? 

4. Shannon’s cat, Poochie, loves boxes. She wants her finance to make a box for Poochie so that Poochie can jump in the box and sit whenever she wants. To make sure Poochie has enough room, the length of the box  be 5 times the width. he is cutting 5 in corners out of the cardboard so he can fold up the sides. he needs to make the box have a volume of 800 cubic in. what are the dimensions of Poochie’s box? round to the nearest inch. 

Answer 1

Hey, no fair including 4 questions in a single "question"...

I'll do #1 though ... the Volume of the folded tin box is given by

V = length * width * depth, which is

(20-2x)(16-2x)x = 320x - 72x² + 4x³

^{Using a graphic calculator, it looks like maximum volume is achieved if x is approximately 2.95, and the volume value at x=2.95 is about 420.1 cubic inches.}

^{If I cheat and use a little elementary calculus, I can get a more-precise value ... x = 2.945 ... by finding the derivative of the Volume function, and solving the resulting quadratic equation 3x^2 - 36x + 80 = 0.}