Question

If 1200cm^2 of material is available to make a box with a square base and an open top, find the largest possible volume of the box

Answer 1

the volume of a box is given by:

V = lwh length times width times height right?

the surface area of this box is equal to 4 sides + a bottom..

As = lw + 2lh + 2wh

Answer 2

we know 1200 is the total surface area so....

1200 = lw +2wh + 2lh

Answer 3

if we can get the value of l,w,and h into single terms wed be doing good :)

Answer 4

labis, since you were told that the base is a square, this can be simplified.

Let x equal the dimension of the base. Now the volume is simply the area of the base times the heigth or \[V=x ^{2}h\]

Answer 5

The surface area is now simplified as it is equal to the area of the base plus the area of the 4 sides:\[A=x ^{2}+4xh\]
This is given to equal 1200 cm. \[x ^{2}+4xh=1200 cmsq\]

Answer 6

From this equation express h in terms of x. This can be done as follows: Subtract x^2 from both sides giving.
4xh=1200-x^2
Now divide through (both sides of equation) by 4x.\[h=(1200-x ^{2})/4x\]

Answer 7

Using standard rules for simplifying fractions, adding, subtracting etc. This becomes:

h= 300/x - x/4

Now plug this value of h into the volume equation above. You then get:
\[V=x ^{2}(300/x - x/4)\]

Answer 8

This reduces to:
\[V =300x-1/4x ^{3}\]

Now differentiate and set to zero\[300-3/4x ^{2}=0\]
Subtract 300 from both sides \[-3/4x ^{2}=-300\]
Multiply both sides by 14/3 getting:\[x ^{2}=400\]

Answer 9

Solve for x getting 20 cm. You now can solve for h getting 10 cm
V is the sum of x squared times h or V=4000 cubic cm.

Answer 10

I hoped you follow with understanding.