# Optimization Problem

• July 21st 2010, 05:25 PM
jpratt
Optimization Problem
If 1700 square centimeters of material is available to make a box with a square base and an open top, find the largest possible volume of the box.
Volume = cubic centimeters

1700 = x^2+4*x*h
4xh=1700-x^2
h=1700-x^2/4x
V=x*(1700-x^2/4)
=1700x-x^3/4
V(x)=1700/4x-1/4x^3
V'(x)=1700/4-3/4x^2
set v'(x)=0
1700/4-3/4x^2=0
3/4x^2=1700/4
3x^2=1700

I am really stuck and i need some guidence and some to compare their answers with me. (Worried)
• July 21st 2010, 06:38 PM
skeeter
Quote:

Originally Posted by jpratt
If 1700 square centimeters of material is available to make a box with a square base and an open top, find the largest possible volume of the box.
Volume = cubic centimeters

1700 = x^2+4*x*h
4xh=1700-x^2
h=1700-x^2/4x
V=x*(1700-x^2/4)
=1700x-x^3/4
V(x)=1700/4x-1/4x^3
V'(x)=1700/4-3/4x^2
set v'(x)=0
1700/4-3/4x^2=0
3/4x^2=1700/4
3x^2=1700

I am really stuck and i need some guidence and some to compare their answers with me. (Worried)

$\displaystyle V = 425x - \frac{x^3}{4}
$

$\displaystyle V' = 425 - \frac{3x^2}{4} = 0$

$\displaystyle x = \sqrt{\frac{1700}{3}}$

this value of x yields a maximum since v'' < 0 for all values of x in the usable domain.