If 12 ft2 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

nathanjhw · Answer

@jim_thompson5910

misty1212 · Answer

HI!!!

misty1212 · Answer

lets call the base $x$ and the height $h$ so that the volume is $V=x^2h$

misty1212 · Answer

then the total area is 
$$x^2+4xh$$ which you know is 12 so set 
$$x^2+4xh=12$$ solve for $h$in terms of $x$ and put it back in the the equation of the volume

misty1212 · Answer

you good from there?

nathanjhw · Answer

I'm still confused.

nathanjhw · Answer

@misty1212

nathanjhw · Answer

I think the answer is 2ft^3 is that correct?

nathanjhw · Answer

@misty1212

misty1212 · Answer

i have no idea

misty1212 · Answer

seem unlikely it is a whole number , but it could be

nathanjhw · Answer

The possibilities are  
        2 ft3
       4 ft3
       5 ft3
       8.5 ft3
       9 ft3

misty1212 · Answer

$$x^2+4xh=12\
4xh=12-x^2\
h=\frac{12-x^2}{4x}$$ so 
$$V(x)=x^2\times\frac{12-x^2}{4x}$$

misty1212 · Answer

clean that up, take the derivative etc

nathanjhw · Answer

When I solved for h and x I got integer solutions h= -1 , x = -2 ; h= -1, x= 6; h=1 , x=-6; h=1 , x =2

nathanjhw · Answer

I plugged it back in and got 2 and 6 and assumed 2 was the answer because it was one of the choices.

nathanjhw · Answer

@misty1212