Question

An open top shipping crate with a square bottom and rectangular sides is to hold 32 inches cubed and requires a minimum amount of cardboard. Find the most economical dimensions.

Answer 1

we can do this if you like, it is not too hard

Answer 2

yeah it would be great if you helped me

Answer 3

ok put the sides of the base as x, so the area of the base is
\[x^2\] and the hight as h so the volume is
\[x^2h\]

Answer 4

and since you know the volume is 32, you know
\[x^2h=32\] so
\[h=\frac{32}{x^2}\] so now we can write the surface area only in terms of x

Answer 5

now the surface area is 4 sides and the bottom. the bottom has area
\[x^2\] and the sides each have area
\[xh\] and there are four of them so the total area is
\[x^2+4xh\] then replace h by
\[\frac{32}{x^2}\] and get the surface area S as a function of x alone

Answer 6

okay i got it from here. thank you

Answer 7

i get
\[S(x)=x^2+4\times \frac{32}{x^2}=x^2+\frac{128}{x}\]

Answer 8

do you think you could help me through 2 more problems?

Answer 9

check my algebra, but if it is right now you job is to minimize the surface area. take the derivative, set it equal to zero, solve, etc.

Answer 10

sure you can ask here or post afresh either way.

Answer 11

damn typo in this line it should have been
\[S(x)=x^2+4x\times \frac{32}{x^2}=x^2+\frac{128}{x}\]