Math Help - Optimization w/ Surface Area

1. Optimization w/ Surface Area

I am given the dimensions of a box (h=14,w=10,l=3) I have to preserve the ratio of H:W, which is 7:5. L does't need to be porportional to anything. I am told I must maintain the H:W ratio and the volume. The volume I found to be 420 in.^3.
The surface area equation is 2lw+2lh+2wh
I need to minimize the amount of surface area on the box while maintaining the specified factors including the volume.
I have tried many things including replacing lw, lh, wh, none have worked. Can someone please show me how to do this, I need to find:
• Equation of SA(l) where SA(l)=aL^2+b/L^c
• The minima

If someone can please show me how to do it?

2. Re: Optimization w/ Surface Area

$LHW = 420$

$W = \frac{5H}{7}$

$LH \cdot \frac{5H}{7} = 420$

$H = \sqrt{\frac{7 \cdot 84}{L}} = 14\sqrt{\frac{3}{L}}$

$W = 10\sqrt{\frac{3}{L}}$

$A = 2(LH+LW+HW)$

$A = 2\left(14\sqrt{3L} + 10\sqrt{3L} + \frac{420}{L^2} \right)$

$A = 24\left(2\sqrt{3L} + \frac{35}{L^2} \right)$