Shipping crates A square-based, box-shaped shipping crate is d...

Question

Shipping crates A square-based, box-shaped shipping crate is designed to have a volume of 16 ft³. The material used to make the base costs twice as much (per square foot) as the material in the sides, and the material used to make the top costs half as much (per square foot) as the material in the sides. What are the dimensions of the crate that minimize the cost of materials?

Accepted Answer

Welcome back, everyone. In this problem, a thermal storage chest with a square base and a box-like shape must hold 20 cubic feet of material. The bottom panel is made of reinforced insulation, which is 3 times more expensive per square foot than the side panels, while the top lid is constructed from a lightweight material that costs the same as the sides. Determine the side lengths of the square base and the height edge of the storage chest that will minimize the total material cost. Now let's try to visualize what's going on here to help us figure out the side length and height that will minimize the material cost. Now, we're told that our chest has a square base, OK, and a box-like shape. So we can imagine that it would probably look something like this, OK? Where on our base, it has the lengths of S. And then it will have a height of age. OK? Now, in this case, if we think about it here, then that means since the volume of the chest is 20 cubic feet from our diagram we can tell that the volume is going to be equal to the cross sectional area Squad multiplied by the height H and that is gonna be equal to 20 cubic feet, OK. Now what else do we know? Well, if we use our volume here, that means our height H, or we can express our height in terms of S because the height then is gonna be equal to 20 divided by Squared. OK? So let's keep this in mind. We are, we, we've not been able to write H in terms of S. What else do we know? Well, we're trying to minimize the total material cost, OK? And we'll need to spend money to get square square footage of our material, OK? No, and we know that, sorry, because we're told that the bottom panel is 3 times more expensive per square foot. So that's why we're interested in our material per square foot. So that means we can find the total material cost if we think about the surface area of our storage chest. Now what do we know? Well, if we think about it here, OK. The surface areas of different parts of the chest. Are the base square, which is Square, OK. The top square, which is also Squad, and then we have four sides which are all rectangles uh of an area S multiplied by H. OK? So, in that case, we can say Uh, the base square. Is squared. The top square Is squared. And our four sides. Equals 4 S H which we know H is 20 divided by squared plus 4s multiplied by 20 divide by squared which equals 80 divided by S N. So now that we have that, let's think about what we know. We're told the base costs 3 times as much per square foot as the sides, the top costs the same per square foot as the sides, and the sides have a base cost of C per square foot. So thus, that means the total cost function. Keeping in mind our square footage. Is going to be CFS equals 3 C squared. Plus C of Squared plus C of 80 divided by S. In that case then, This means the total cost is going to be C of 3S2 plus S2 + 80 divided by S, which is going to be equal to C of 4S2 plus 80 divided by S. This is our cost function. Now, to minimize the total cost, we can differentiate C and set it to 0 to get a critical point, and then we can just test to make sure that it is the minimum point. OK? So let's differentiate C of S. And set to 0 and setting it to 0. OK. Then We know here we can differentiate these terms using the poor rule. So the derivative of CFS. It's going to be equal to. C of 8 S minus 80 divided by Square, and this is what is gonna be equal to 0. No, this tells us then, OK, that if we combine here both of these terms. Then it is cubed. OK. It is cubed minus 80 divided by Squared is going to be equal to 0. Once Squad is not 0, OK? And thus, if we come over to the left here, that means uh If we multiply through by Squad, 8 is cubed minus 80 equals 0, so 8 is cubed equals 80, divide through by 8 is cubed equals 10. So S equals the cube root of 10. Feet Now that we have our critical value for S, is it the minimum cost? Well, we can find that with the help of the second derivative of the cost function. OK. So solving for The second derivative of our cast function. To use the second derivative test. This means that we're going to differentiate, uh, we're gonna differentiate 8C of 8 S minus 80 divided by Squared again. And when we do that, OK, then we should get our second derivative to be equal to C of 8 + 160 divided by SQ, OK, where C is greater than 0 because it represents the material cost. Now, what do we know about the nature of our second derivative here just by looking at it? Well, we can tell that for all positive values of S, OK, then the second derivative is also going to be positive, OK. Notice where S is dividing by Scubed. Therefore, Because our second derivative is positive, that means the function is concave up, and once it's concave up, that means. At S equals the cube root of 10 ft. Gives the minimum cost. Gives not it gives the minimum cost, OK. So To sum up what we just did, we know that the second derivative is positive, so that means it gives the minimum cost. Know that we have that, we can solve for H because we already know that S is gonna be the cube root of 10 ft. So remember, We said that H was equal to 20 divided by Squad. Since we have a value for S, now we know it's gonna be 20 divided by the cube root of 10 squad. When we solve here, OK, uh, no, that's gonna be 20 divided by the cube root of 10 squad, which we know here if we can uh find the conjugate here just to simplify. So that's we're gonna multiply by the cube root of 10 in the numerator and denominator. OK. And then when we do that, OK, we get 20 cube root of 10 divided by 10, OK. And 20 divided by 10 is 2. So H equals 2 multiplied by the cube root of 10 ft. Thus That tells us then for us to minimize the total material cost, the side length of our storage chest needs to be the cube root of 10 ft and its height has to be 2 multiplied by the cube root of 10 ft. Thanks a lot for watching everyone. I hope this video helped.

Key Concepts

Volume of a Box

Cost Function

Optimization Techniques

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