Optimization: Minimize Surface Area Of A Box Given The Volume
All right another optimization problem have quite a few of them for you to look at you'll. You can search my channel, or you can find them in the derivatives playlist. So here we have a box with a square. Now that should say a square base, an open-top must have a volume of 32,000 cubic centimeters, find the dimensions of the box that minimize the amount of materials used all right so let's draw our box here. All right. So we have a square base. Remember that should say base and so well they will call the.
Base X and X, I know, they're the same because it's square, and then we have the height. And it tells me the volume needs to be 32,000 all right. So that tells me that 32,000 is equal to x times, X times, H that's, the volume of a box. Okay. So x times X is x squared times.
The height H all right so that's, some information that they gave us. Now. What do they want us to in this case minimize they want us to minimize the materials used? So they want us to minimize the surface area. Okay. So let's get a function. For the surface area.
So that's, just the area of each face added together. So I've got the area of the bottom here, which is x times X so that's x squared, and then I need the surface area of this side. This side, the front and then the back. Well, the area of this is x times H, right all right so let's, look at that x times H. Well, you see this height here, that's H, this is X. So that area is x times H. So I've got two of them and then let's look at the front. Well, this is X. This is H. So this area is x times.
H and then the same thing with the back that's X back there, and this is H, so actually I've got four of them so that's going to be 4x H. Now we need to take the derivative of this set it equals to zero well. But before we do that I want to get everything in terms of one variable. So let's get everything in terms of X, so we'll solve this over here for H. So you can see here, I know that H is 32,000 over x squared. So I'll plug this in for H. So I get s is x squared plus 4x times 32,000 over x squared. So. That's going to give me the surface area is x squared, plus, you know, let's see that's going to be 128 thousand over X, the four times 32,000.
And then this X and this X cancel all right. So now I need to do what take the derivative. So the derivative is 2x, minus 128 thousand over x squared. Now, remember, when we take the derivative of this let's, just look real quick over here to the side 128, and that would be X to the negative 1 right move that up so that's negative 128. And then 28,000 subtract 1. Negative 2 move that back down to the denominator and that's where this comes from all right. So now let's set this equal to zero.
So that's, 2x, minus 128 thousand over x squared equals zero. So that gives me 2x equals 128 thousand over x squared. And so that gets moved up. I, get that's X cubed equals. And then when this moves across the two that goes to the denominator.
And so it'd be a hundred and twenty-eight thousand divided by two, which that would give me 64,000. And so I get X it's. The cube root of. This right axis the cube root.
So that actually comes out to forty centimeters. So that's X. And what do they want to know? You know, they're one of the dimensions of the box.
So the dimensions of the box? Well, there's X. And now we need the height H. Well, here's H. So H is equal to 32,000 over 40 squared. So H comes out to be 20 centimeters.
So the dimensions is 40 by 40 by 20. So I hope, the video helped check out my other videos, give me a like share and subscribe. And thanks for watching.