Applied Maxima and Minima Problems

[courtrigrad]

Hello all

I have a few questions on applied maxima and minima

1. A company gives you 675 sq. ft of cardboard to construct a rectangular carton with the largest volume. If the carton is to have a square base and an open top, what dimensions would you use?

My Thought Process:

Volume = length * width * height
= x*x* (675- 2x)

I know how to find the maximum volume, but am not sure about the actual equation. If it has a square base, then shouldn't it be x*x *( 675 - 2x)


Any help is greatly appreciated

Thanks

 

[HallsofIvy]

It's always a good idea to specify WHAT your symbols represent. Is x a length? I assume that it is the length of the sides of the square bottom, in feet.
But if that is the case then "675- 2x" makes no sense. 675 square feet is total surface area of the box and 2x has units of feet. You can't subtract feet from square feet!

Your basic idea is right: If you call the lengths of the sides x and the height y, you have one side with area x² square feet and 4 sides with area xy square feet. The total area is 675. Use that to find y.

 

[courtrigrad]

thanks a lot

just to clarify, after finding y I would then just substitute this back into the volume equation and then find maximum

 

[arildno]

"2. A printed textbook page must contain 20 sq. in. of printed matter with a 2 inch margin on each side and at the top and with a 3 inch margin at the bottom. What dimensions must the pages have in order to minimize the amount of paper used?

I know the dimensions of the paper must be x(length) and 10 - x(width) How would I incorporate the information given about the margins?"

Why do you think this?
Let L be the horizontal length of the paper, H the height of the paper, x the length of the printed matter, y the height of printed matter.
We therefore have the equations:
2+2+x=L
3+2+y=H
xy=20

And you are to minimize the product LH
Hint, solve for y, and write LH as a function of x to be minimized.

 

[courtrigrad]

thanks a lot everyone.

for # 4 and # 5 can someone give me a hint in how to get the function?

Thanks