Question

A manufacturer wants to design an open box (no top) having a square base and
a surface area of 300 square inches. What dimensions will produce a box with
maximum volume?

Answer 1

I don't even know where to start on this. :(

Answer 2

thats not right. I know i have to maximize it but just don't know how.

Answer 3

it is using derivative

Answer 4

normally, if the question wasn't so abstract I'd get it. I would probably find Surface area, then take derivative and plug in one of my variable to eliminate then do a test to see if max.

I just feel like its missing some numbers, but I know its not because I know there is an answer to it...kinda remember seeing something like this in class, but didn't get it.

Answer 5

I'm use to already being given the equation lol. but this one doesn't give it to me.

Answer 6

the equation is: surface area = 300 = b^2+4bh

Answer 7

The answer for the box dimension to give maximum volume and a surface area of 300 sq. ins. is :
Height = 5 inches
Width = 10 inches
Depth = 10 inches
Posting of solution method to follow.

Answer 8

patiently waiting.

Answer 9

surface area of the box is= b^2+4bh=300. volume of box is V=b^2h.
h=V/b^2 => S=b^2+ 4V/b=300. => V=(300b-b^3)/4. => dV=(300-3b^2)/4=0
=> b=10. substitute this in surface area equation to get height to be 5 inches.

Answer 10

Let base side = b inches
Let height = h
Area of sides = 300 - b^2
300 - b^2 = 4 * b * h
\[h=\frac{300-b ^{2}}{4b}\]
\[Volume=b ^{2}(\frac{300-b ^{2}}{4b})\]
\[V=75b-\frac{b ^{3}}{4}\]
\[\frac{dV}{db}=75-\frac{3b ^{2}}{4}\]
Putting f'(b) = 0 and solving gives b = 10
Substitution gives h = 5

Answer 11

Thank you everyone for all your help. I understand now. I wish I could give out more than 1 medal because you all contributed to helping me find the answer.