Optimization: Minimum Surface Area

[JoeyC2488]

Homework Statement

I need help on an optimization problem involving a hexagonal prism with no bottom or top, but the top is covered by a trihedral pyramid which has a displacement, x, such that the surface area of the object is at a minimum for a given volume. The assigned variables include:
A= the surface area of the object
a= the length of each side of the hexagon (a=1)
h= the height of the prism
x= displacement

Homework Equations

Given that A= 6(ah-1/2ax) + 3a\sqrt{3}*\sqrt{x^2+(a^2/4)}, use calculus to find the displacement, x that yields the minimum surface area. Calculate x if a=1.

The Attempt at a Solution

I'm struggling trying to find a secondary equation for this optimization problem and I think I need to use an equation evolving h, but I'm not sure what.

 

[arildno]

The volume V is to be constant, this will lead you to an equation in h and x, from which you may find an expression for h.

 

[JoeyC2488]

How would I use a formula for volume since the problem works with two shapes.

V=(3\sqrt{3})/2 a^2*h

h= V/ (3\sqrt{3}/2 a^2*h

Then, I can just plug this into the first equation, simplify, and start the calculus. Is that right?

 

[arildno]

Your TOTAL volume is the sum of the prism's volume and the pyramid's volume.

That is the equation in h and x you are to set up!

 

[arildno]

Your total volum is therefore:
V=\frac{3\sqrt{3}}{2}a^{2}h+\frac{1}{3}\frac{3\sqrt{3}}{2}a^{2}x\to{h}=\frac{2V}{3\sqrt{3}a^{2}}-\frac{x}{3}

 

[JoeyC2488]

So after substituting that in for h, the simplified form would be:

A= 4V-5x\sqrt{3}+ 9\sqrt{x^2+4}

\sqrt{3}

 

[JoeyC2488]

How would I take the derivative of this equation then with multiple variables?

 

[arildno]

You only have one variable, x. Remember, V is a constant!