Posted by **Anonymous** on Sunday, November 29, 2009 at 10:13pm.

If 1800 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=____________

I did sqrt(1800)=42.4264

and then 42.4264/3= 14.142 cm

then 14.142^(3) and got volume= 2828.34575

however is was incorrect...please help me figure out what I'm doing wrong

THANK YOU!

- Math!! -
**Damon**, Sunday, November 29, 2009 at 10:38pm
height = h

side of bottom = s

area of bottom = s^2

area of each of 4 sides = sh

so total area = s^2 + 4sh

1800 = s^2 + 4sh

h = (1800-s^2)/4s

volume = v = s^2 h

so

v = s^2 (1800-s^2)/4s

v = s(1800-s^2)/4

4v = 1800 s - s^3

where is that maximum?

by trial and error I get s = 24.5

then h = .5

volume = 300

- Math!! -
**MathMate**, Sunday, November 29, 2009 at 10:39pm
The box is made up of a square base of side x, and 4 sides each of height h and width x.

The total area is therefore

A=x² + 4hx

Since the area is known, h can be expressed in terms of the area A

A=x²+4hx

or h=(A-x²)/4x

The volume, V is given by

V(x)=x²h

=x²*(A-x²)/4x

=x(A-x²)/4

Use your knowledge of calculus to find

V'(x), and if there is a maximum, the value of x can be found by equating the derivative to zero,

V'(x)=0.

Solve for x and hence V.

I get about 7300 cm³

- Math!! -
**Anonymous**, Sunday, November 29, 2009 at 10:52pm
Thanks both of you guys for attempting this problem, and a special thanks to MathMate! The answer was about 7248 cubic cm...

Thanks!!

- Math!! -
**MathMate**, Sunday, November 29, 2009 at 11:14pm
For x I got 10√6=24.495.

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