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 $V(x)=2 x^{2}(18-x)$ (b) What is the domain of $V ?$ (Use the fact that length and volume must be positive.) (c) Draw a graph of the function $V$ and use it to estimate the maximum volume for such a box. (FIGURE CAN'T COPY)

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 $V(x)=2 x^{2}(18-x)$
(b) What is the domain of $V ?$ (Use the fact that length and volume must be positive.)
(c) Draw a graph of the function $V$ and use it to estimate the maximum volume for such a box.
(FIGURE CAN'T COPY)

Key Concepts

Graphical Analysis and Optimization

Graphing a function is a powerful tool to visualize its behavior, including identifying local or global extremes. This technique is essential for optimization problems, where one seeks to determine the maximum or minimum value of a quantity subject to given constraints.

Function Domain in Context

The concept of a function’s domain refers to all possible input values that make sense in a given situation. In physical problems like this, the domain is limited by real-world constraints such as the requirement for lengths and volumes to be positive.

Modeling with Functions

In many real-world scenarios, relationships among quantities are expressed using functions. Here, constraints such as fixed edge lengths lead to an expression for the volume as a function of one variable, encapsulating the relationship between the box’s dimensions and its volume.

Volume Calculation of a Rectangular Prism

This concept involves computing the volume of a three?dimensional box by multiplying the area of the base by the height. When the dimensions are expressed in terms of variables, it results in a function that describes how the volume changes as the dimensions of the box change.

Transcript

00:01 All right, so here's our question.

00:02 We're finding the volume of a cardboard box that has a square base.

00:05 It is not the best drawing of a cardboard box i've made in my life, but it definitely has a square base that is x by x long.

00:13 So here's what we know.

00:14 We know the total length of all 12 edges is 144 inches.

00:20 And we have to show that the volume of the boxes should really be written up here is represented by 2x squared parentheses 18 minus x, which is kind of odd because there's a dimension here.

00:32 I'm going to call it y that we don't know.

00:34 So it's kind of odd to me that we can write the volume of this box without having another variable that we don't know.

00:42 But here's why that is.

00:44 Let's start with the fact that we know that 144 inches is the total sum of all 12 edges.

00:51 Well, i know that down here, if i'd maybe make a dotted line, i know that these four edges are all x.

00:59 But guess what? if i go up here, these four edges at the top are also square and also x.

01:06 So there's eight xs there, which really leaves me with the only edges i haven't counted.

01:12 Maybe i'll use another color are these kind of tall edges.

01:16 Or at least they're tall in our drawing, in our picture.

01:21 But let's call those y, right? so let's say the sum of them is 8x plus 4y.

01:28 And let's just use some algebra and solve for y.

01:30 So we could do 144 minus 8x.

01:36 I'd get this by subtracting 8x from both sides is equal to 4y.

01:41 Now if i multiply both sides by 1 fourth or divide by 4, that's going to put both of these terms over 4.

01:48 So 144 over 4 minus 8.

01:55 I'll just rise the coefficient over 4x.

01:59 And then 4 over 4 is 1, so i'd just have y.

02:02 Well, let's put that next step up here because we ran out of room.

02:05 144 divided by 4, that's 36, minus 8 divided by 4 is 2 is our y.

02:14 And if i wanted to go one step further, i could factor out a 2 and say, okay, that leaves me with an 18 here because 2 times 18 is 36.

02:23 2 times 2, well, obviously, or if i pull a 2 out of a 2x, i got it.

02:28 So that just leaves me with the x and that's equal to y.

02:33 Last thing.

02:34 Let's talk about volume.

02:36 If our volume is just going to be length times width times height, well, that's x times x, so x squared times y.

02:46 But wait a second, we know what y is.

02:48 We just solved for it...

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