Optimization Problem

Jun 2010
21
0
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)
 

skeeter

MHF Helper
Jun 2008
16,217
6,765
North Texas
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)

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

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

\(\displaystyle \displaystyle x = \sqrt{\frac{1700}{3}}\)

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