[Technology Exercise] 17. Designing a suitcase A 24-in.-by-36-...

Question

[Technology Exercise] 17. Designing a suitcase A 24-in.-by-36-in. sheet of cardboard is folded in half to form a 24-in.-by-18-in. rectangle as shown in the accompanying figure. Then four congruent squares of side length x are cut from the corners of the folded rectangle. The sheet is unfolded, and the six tabs are folded up to form a box with sides and a lid.

b. Find the domain of V for the problem situation and graph V over this domain.

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b. Find the domain of V for the problem situation and graph V over this domain.

Accepted Answer

Hello. In this video, we are told that a bakery is designing a custom cake box using a 20 inch by 30-inch sheet of cardboard. The cardboard is folded in half to form a 20 inch by 15 inch rectangle. To create the sides of the box, 4 identical squares with side lengths 10 inches are cut from each corner of the folded rectangle. The cardboard is then unfolded, and the 6 flaps formed by the cuts are folded upward to create the sides of the lid of the box. Let V of X represent the volume of the finished cake box to determine in terms of X. We want to determine the domain of V. So, we're learning, we want to find some function V of X. That defines the volume of the box. Now, the volume of the box is defined as the length times the width times the height. So, if we take a look at the given image, notice that we have this base of the box that has a length of 20 inches and a height of 30 inches. Now, each of these side flaps here have a length of X. That is going to be X in each corner. So, we can use this to first determine the length. Now, for the length of the box, it is going to be the entirety of the 20 inches minus the length of these two flaps that we cut out. So we can define length to be 20 minus 2 X. Now what about the width of the box? Now, we don't know what the width of the box is going to be, but we can just label the width as two sides of X. So width is going to equal to 2 X. Now finally, what about the height? Now, the height of the box is defined as these 30 inches, but just like before, we are going to determine the height of the box by first subtracting 2 X from the length of the of the height. So height of the box is going to be 30 minus 2X, and because there's two sides to the height, we are going to divide this value by 2. This is going to allow us to simplify the height to be 15 minus 2X. Now that we've determined the length, width, and the height, we can rewrite the function V of X to be V X is equal to the quantity, 20 minus 2 X multiplied by 2 X, multiplied by 15 minus 2 X. And now what we want to do is we want to determine the domain in terms of X. Now, because we are dealing with distances, X has to be a positive number greater than 0. So in order to determine what the minimum values of X are going to be, we are going to take each piece of the function and and set it equal or greater than 0. So, 20 minus 2 X must be greater than 02 X must be greater than 0, and 15 minus 2 X must be greater than 0. Now for the first inequality, when we solve for X, we get that X must be less than 10. For the second inequality, solely for X gives us that X is greater than 0. And for the final inequality, we get that X is less than 15 divided by 2. Now, we know that the minimum value of X is going to be zero since we're dealing with distances, so the domain is going to have a minimum value of 0. But when picking the maximum value, we want to pick the most restrictive value for X, and in this case here, 15 divided by 2 will be smaller than 10, because the approximate value of 15 divided by 2 is 7.5. So the maximum value that X can be is going to be 15 divided by 2, and this is going to be the domain of the volume function. So I hope this video helps you in understanding on how to approach this problem, and we will go ahead and see you all in the next video.

Key Concepts

Volume of a Box

Domain of a Function

Graphing Functions

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