A manufacturer wants to design an open box having a square base and a surface area of 216 square inches. What dimensions will produce a box with maximum volume?

This is what I got when I solved it and is it correct?

S= (area of base)+ (area of four sides)
S=x^2 + 4xh = 216

=x^2 (216-x^2/4x)= 54x- x^3/4

0< x < square root 216

dV/dx = 54- 3x^2= 0

then 3x^2 = 216 (when divided it's 72)

x= -9, -8

The lines
S= (area of base)+ (area of four sides)
S=x^2 + 4xh = 216

all look OK.

Solving for h gives h = (216 - x^2)/(4x) { note the use of parenthesis is necessary }.

The next line would then be h = 54/x - x/4.

This can be put into V = x^2 h giving V(x) = x^2(54/x - x/4).
That works out to V(x) = 54x - x^3 /4.
This then give dV/dx = 54 - 3x^2 /4.

Solving give 3x^2 = 216.

Yes, this works out at this, but things weren't written correctly in between.
Also, the solution wasn't found correctly.

Now if 3x^2 = 216, then x^2 = 72, so x = 6*sqrt(2).
That can be put into h = 54/x - x/4 to find h.
Once this has been done, put both x and h into V = 2 x^2 h.


All Answers

Answers by Expert:

Ask Experts




Any kind of calculus question you want. I also have answered some questions in Physics (mass, momentum, falling bodies), Chemistry (charge, reactions, symbols, molecules), and Biology (reproduction, insusion of chemicals into bloodstream).


Experience in the area: I have tutored students in all areas of mathematics since 1980. Education/Credentials: BSand MS in Mathematics from Oregon State University, where I completed sophomore course in Physics and Chemistry. I received both degrees with high honors. Awards and Honors: I have passed Actuarial tests 100, 110, and 135.

Maybe not a publication, but I have respond to well oveer 8,500 questions on the PC. Well over 2,000 of them have been in calculus.

I aquired well over 40 hours of upper division courses. This was well over the number that were required. I graduated with honors in both my BS and MS degree from Oregon State University. I was allowed to jump into a few courses at college a year early.

Awards and Honors
I have been nominated as the expert of the month several times. All of my scores right now are at least a 9.8 average (out of 10).

Past/Present Clients
My past clients have been students at OSU, students at the college in South Seattle, referals from a company, friends and aquantenances, people from my church, and people like you from all over the world.

©2017 All rights reserved.