Volume of a box a cardboard box has a square base with each...

Volume of a Box A cardboard box has a square base, with each edge of the base having length $x$ inches, as shown in the figure. The total length of all 12 edges of the box is 144 in. (a) Show that the volume of the box is given by the function (b) What is the domain of $V ?$ (Use the fact that length and volume must be positive.)$V(x)=2 x^{2}(18-x)$ (c) Draw a graph of the function $V$ and use it to estimate the maximum volume for such a box.

Transcript

00:02 Okay, in this activity we're told that we have a box with a square base, which means that all of the sides of the base are equal.

00:13 We don't know what that base is, but we're going to call it x.

00:17 And we are told that the total length of all of the sides added together is 144 inches.

00:35 Double -checking that i've got the right units.

00:39 So i -144 inches total edges and we're supposed to find an expression for the volume in terms of the length of side x.

00:58 So we need to figure out first of all how can i express the height of the box in terms of the length x.

01:12 Well, i know that the total length of all of the sides is 144.

01:18 There are 12 sides.

01:19 One, two, three, four on the top, one, two, three, four on the base, and then one, two, three, four vertical edges.

01:31 So all of those 12 sides total up to 144 inches in length.

01:37 So i'm going to look and see i have eight edges.

01:46 Edges that are x units, x inches in length, the four on the top and the four on the bottom.

01:55 And then i've got another four.

01:57 Oops, i don't know why i wrote an equal sign because i am talking while i am trying to write.

02:04 I am going to add four more edges that are the height and that has to add up to 144.

02:16 The problem is now i've got two variables.

02:18 I've got x and the height.

02:20 I want to find a way to rewrite the height in terms of the length of one side of the base x.

02:28 So i'm going to solve this equation for h so that i can use that expression, x, instead of h.

02:40 Let me show you what i mean.

02:41 I'm going to start by dividing everything by 4 just to make this easier.

02:45 8 divides by 4.

02:46 4, 4 divides by 4, 144 divides by 4, and i get 2x equals h.

02:53 Oops, 2x plus h equals 144 divided by 4 is 56.

03:07 No, sorry, 36.

03:10 See what you get for trying to do math in your head? if you put this in a calculator, you get 36 inches.

03:19 And then subtracting 2x from both sides.

03:23 That the height is 36 inches minus 2x.

03:29 I can also factor a 2 out of this and say that the height is equal to 2 times 18 minus x.

03:49 So this is the height of the box based on the length of one side of the base.

03:58 And volume of the box, of course, is length times width times height.

04:07 We know that the length and the width are both x, because we're told that the base is square.

04:15 And we now know that the height is 2 times 18 minus x.

04:25 And if i rewrite that a little bit and multiply the two out front, that means that the volume is 2x squared times 18 minus x.

04:41 So that is an expression for the volume.

04:43 Then i'm asked to find what is the domain, which means what are all of the possible x values for this situation? well, they want us to consider that i'm not going to have a negative length.

04:59 So the length of each of the sides of the base has to be greater than zero.

05:04 It has to give a positive volume.

05:08 I'm not going to have a negative volume...

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