Optimization Problems

[Jess]

I absolutely hate optimization problems. For some reason, I get the ones that are given as examples in class, but on tests and homework I don't get them at all. I pretty much failed on this lesson in Calc last year ....totally killed it on a test on it.

Here are four that I need to do. Please help me out x_x

1. 4904 SOLVED

2. 4903 SOLVED, thank you Gigablue

3. 4902

4. 4901


I need the answers by 11:45 pm EST!!!

UPDATE: SOLVED FIRST PROBLEM! still need help with the other 3. Thanks in advance

[Gigablue]

For number two:

V=lwh, l=7-2x, w=10-2x, h=x
V=(7-2x)(10-2x)(x)
V=4x^3-34x^2+70x
V'=12x^2-68x+70

Let V'=0
0=12x^2-68x+70
0=6x^2-34x+35

After applying the quadratic formula
x=(17+sqrt79)/6, x=(17-sqrt79)/6

The first answer is in admissible
Therefore x=(17-sqrt79)/6

[Sonic Boom]

3. length = 2w; width= w; height = h

So Volume= 71 = 2hw^2

Cost = ($3 x 2w^2) + $3[2wh + 2(2wh)]
= 6w^2 + 6wh + 12wh
=6w^2 + 18wh

To minimize cost, set dC/dw = 0 = 12w +18h

=> 18h = -12w
=> w= -3/2h and so length = -3/h and height = h

But Volume = 71! ....So (-3/2h) x (-3/h) x h = 71 or 71= 9/2h

=> 2h = 9/71
=> h= 9/142

Then substitute this back into the original equation: 71 = 2w^2 x 9/142

....and then you can find w and 2w by using your calculator.

This may have a mistake in it tho (it's late over here). So just double check to see if I made any errors in the steps.