Question

For an open-topped rectangular box, with a square base X by X cm and height h cm, find the dimensions giving the maximum volume, given that the surface area is 40cm6^2

Answer 1

40cm^2

Answer 2

I am not 100% sure the answer, but here is what I think.



\[V = h * x ^{2}\]

Volumn = hight * area of bottom

\[S _{A} = 4hx + x ^{2}\]

Surface area = 4 sides + bottom

There are 2 variable we can use to take derivative .
I choose x.

so we have :
\[S _{A}\prime = 4h + 4h _{x}\prime + 2x = \left( 40 \right)\prime = 0\]

tidy up the equation we get:
\[h _{x}\prime = \frac{ -x - 2h }{ 2x }\]

Now we do the same thing to the volume :
\[V _{x}\prime = h _{x}\prime x ^{2} + 2xh\]

we get V max when the above equation at critical point.
So we need to know when the above equation equal 0.
Put hx ' in the Vx' equation and make it equal to 0.
we get the relationship between x and h.

2h = x

plug this back to the surface equation. you can get the actual value of x and h.