Maximum Volume An open box of maximum volume is to be made...

Maximum Volume An open box of maximum volume is to be made from a square piece of material 24 centimeters on a side by cutting equal squares from the corners and turning up the sides (see figure). (a) The table shows the volumes $V$ (in cubic centimeters) of the box for various heights $x$ (in centimeters). Use the table to estimate the maximum volume. $$ \begin{array}{|c|c|c|c|c|c|c|}\hline \text { Height, } & {1} & {2} & {3} & {4} & {5} & {6} \\ \hline \text { Volume, } V & {484} & {800} & {972} & {1024} & {980} & {864} \\ \hline\end{array} $$ (b) Plot the points $(x, V)$ from the table in part (a). Does the relation defined by the ordered pairs represent $V$ as a function of $x ?$ (c) Given that $V$ is a function of $x,$ write the function and determine its domain.

Maximum Volume An open box of maximum
volume is to be made from a square piece of material
24 centimeters on a side by cutting equal squares from
the corners and turning up the sides (see figure).
(a) The table shows the volumes $V$ (in cubic
centimeters) of the box for various heights $x$ (in
centimeters). Use the table to estimate the
maximum volume.
$$
\begin{array}{|c|c|c|c|c|c|c|}\hline \text { Height, } & {1} & {2} & {3} & {4} & {5} & {6} \\ \hline \text { Volume, } V & {484} & {800} & {972} & {1024} & {980} & {864} \\ \hline\end{array}
$$
(b) Plot the points $(x, V)$ from the table in part (a). Does
the relation defined by the ordered pairs represent $V$
as a function of $x ?$
(c) Given that $V$ is a function of $x,$ write the function
and determine its domain.

Key Concepts

Mathematical Modeling

Mathematical modeling translates a real-world problem into a mathematical formulation. By expressing the volume in terms of the cut size, the problem is modeled by an algebraic function, which aids in analyzing and solving the problem through established mathematical methods.

Graphing and Data Estimation

Graphing involves plotting points to visually represent the relationship between variables. In this problem, plotting the volume against the cut size allows one to estimate the maximum volume and verify that the plotted points form a function, as each x-value corresponds to one volume measurement.

Function

A function is a relation in which every input is associated with exactly one output. Here, the volume of the box is expressed as a function of the cut size, meaning each chosen value of the cut size leads to a unique box volume, which is fundamental to modeling the scenario mathematically.

Optimization

Optimization involves finding the maximum or minimum values of a quantity under given conditions. In this context, the goal is to determine the maximum volume of a box by varying the height of the cut squares, embodying the process of identifying the optimal value in a practical problem.

Domain of a Function

The domain of a function consists of all possible input values for which the function is defined. In physical problems like this one, the domain is determined by practical constraints (such as the dimensions of the material), ensuring that all considered values produce meaningful and physically possible results.

Transcript

00:02 Or ask to answer a question about maximum volume using functions.

00:07 Hold that an open box of maximum volume is being made from a square piece of material 24 centimeters on a side by cutting equal squares from the corners and turning up the sides.

00:20 And this is depicted in a figure following exercise 61 of this section.

00:28 Now in part a, we're told that we're given a table.

00:32 This table shows the volumes v of the box for various heights x.

00:39 We're asking you the table to estimate the maximum volume.

00:48 So in this table, we have two rows.

00:50 First row is the height x and the second row is the volume v.

00:56 Now, we see the height ranges from 1 to 6, and the volumes range from as low as 484 to as high as 1 .4.

01:15 So we might estimate the maximum volume is approximately that entry, 1 ,024, and the volume is in cubic centimeters.

01:44 That's part a.

01:46 Then in part b, we have to plot the points x, v from the table in part a, and determine if the relation defined by the ordered pairs represents v as a function of x.

02:02 Six points to plot.

02:04 This is just going to be a rough sketch.

02:09 So we have our x and y axes.

02:14 Both of them we're just going to include the positive axis.

02:21 So we have a point at one and then about 484.

02:34 So let's say about here.

02:37 Also a point at two and eight hundred.

02:40 Let's say about here.

02:43 A point at nine hundred three nine hundred seven two.

02:47 So a little bit higher up about here.

02:51 A point at 4, 1 ,024, even more higher up, say about here...