# Cuboid tank help

• Nov 28th 2009, 05:55 PM
user_5
Cuboid tank help
A cuboid tank is open at the top and the internal dimensions of its base are x m and 2x m.
The height is h m.
The volume of the tank is V cubic metres and the volume is fixed. Let S m² denote the internal surface area of the tank.

S in terms of x and h
$S=2(2xh)+2x(x)+2(hx)$
$S=2x^2+6xh$

S in terms of V and x
$V=2x * x * h$

$V=2x^2h$

$h=\frac{V}{2x^2}$

$S=2x^2+6xh$

Sub $h=\frac{V}{2x^2}$

$S=2x^2+6x\left(\frac{V}{2x^2}\right)$

$S=2x^2+\frac{3V}{x}$

The maximal domain of S in terms of V and x is $(0,\infty)$

If $2 find the maximum value of S if $V = 1000m^3$

Here's a visual representation of the cuboid
• Nov 28th 2009, 06:02 PM
skeeter
Quote:

Originally Posted by user_5
A cuboid tank is open at the top and the internal dimensions of its base are x m and 2x m.
The height is h m.
The volume of the tank is V cubic metres and the volume is fixed. Let S m² denote the internal surface area of the tank.

S in terms of x and h
$S=2(2xh)+2x(x)+2(hx)$
$S=2x^2+6xh$

S in terms of V and x
$V=2x * x * h$

$V=2x^2h$

$h=\frac{V}{2x^2}$

$S=2x^2+6xh$

Sub $h=\frac{V}{2x^2}$

$S=2x^2+6x\left(\frac{V}{2x^2}\right)$

$S=2x^2+\frac{3V}{x}$

The maximal domain of S in terms of V and x is $(0,\infty)$

If $2 find the maximum value of S if $V = 1000m^3$

using calculus, find $\frac{dS}{dx}$ , determine the critical value and confirm that it is a value of x that minimizes S