00:01
We are asked to construct a rectangular box with a volume 216 cubic inches, with an open top, and which has the minimum possible surface area.
00:12
First let's write down the formula for the surface area.
00:16
The surface area consists from the surface of the bottom, which is x squared, plus the surfaces of the lateral sides, which is x multiplied by h, each lateral side.
00:29
But since there are four, we need to multiply that by four.
00:34
And there is no top, so we are not adding the surface of the top.
00:38
This is a function we want to minimize.
00:42
We want to rewrite that as a function of one variable.
00:45
To do that, we'll recall that the volume is simply the product of all dimensions, so it's going to be x squared h.
00:53
On the other hand, the volume is 216.
00:56
And now we'll use that equation to solve for h.
01:00
From that equation, we'll get that h equals 216 divided by x squared.
01:08
And now we'll plug in that expression for h in the surface area formula.
01:13
We'll get s equals x squared plus 4x multiplied by 216 divided by x squared.
01:25
And that simplifies to x squared plus 216 times 4 equals 864 divided by x.
01:39
Now it's a function of one variable, and we know how to minimize functions of one variable.
01:44
We need to solve the equation s prime equals zero, where s prime is the derivative and equals 2x minus 864 divided by x squared.
01:57
And we want that to be zero.
02:00
Let's bring the expression to the common denominator.
02:04
We'll get 2x cubed minus 864 over x squared equals zero.
02:14
And that means 2x cubed minus 864 must be zero.
02:22
That means x cubed minus 432 is zero.
02:29
Or x cubed equals 432...