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

Question

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!

Answer 1

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

Answer 2

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

Answer 3

Thanks both of you guys for attempting this problem, and a special thanks to MathMate! The answer was about 7248 cubic cm...
Thanks!!

Answer 4

For x I got 10√6=24.495.