Question

A solid rectangular block has a square base .find its maximum volume if the sum of the height and any one side of the base is 12cm ?

Answer 1

`256`

Explanation:

We have to maximize

`V=a^2h` under the condition `h+a=12`

`V=V(a)=a^2*(12-a)=12a^2-a^3`
Differentiating this formula with respect to `a` we obtain

`V'(a)=24a-3a^2`
and again

`V''(a)=24-6a`
`V'(a)=0` if `a=0` or `a=8`
`a=0` is senseless, for `a=8` we get
`V''(8)=24-48=-24<0`
`V(0)=V(12)=0`

`V_max=64*4=256`