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

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) Write the volume $V$ as a function of $x,$ the length of the corner squares. What is the domain of the function? (b) Use a graphing utility to graph the volume function and approximate the dimensions of the box that yield a maximum volume. (c) Use the table feature of a graphing utility to verify your answer in part (b). (The first two rows of the table are shown.)

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) Write the volume $V$ as a function of $x,$ the length of the corner squares. What is the domain of the function?
(b) Use a graphing utility to graph the volume function and approximate the dimensions of the box that yield a maximum volume.
(c) Use the table feature of a graphing utility to verify your answer in part (b). (The first two rows of the table are shown.)

Key Concepts

Function Modeling

This involves setting up a mathematical function that represents a real-world situation. In optimization problems, a function is often derived from geometric relationships to quantify a desired property, such as volume, cost, or area, in terms of a variable that can be manipulated.

Domain of a Function

The domain of a function is the set of all possible input values for which the function is valid. When modeling a physical situation, the domain is typically restricted by practical constraints, ensuring that all quantities remain meaningful and within realistic limits.

Optimization

Optimization is the process of finding the highest or lowest value of a function, commonly to maximize or minimize a desired quantity. This often involves determining critical points where the rate of change is zero or undefined and then deciding which of these points gives an optimal solution under the given constraints.

Graphical Analysis

Graphical analysis is a technique that involves plotting a function to visually inspect its behavior. This method helps in identifying key features such as local and global extrema, trends, and behavior over the function's domain, which is particularly useful in optimization problems.

Verification Using Tabular Data

Verification using tabular data involves generating a table of input-output pairs for a function. This technique is used to confirm the behavior observed in the graph, validate approximate solutions, and ensure that the numerical findings are consistent with the observed trends.

Transcript

00:06 In this question, we're given a square box, something that looks like this.

00:16 And we're told that the side is 24 minus 2x.

00:23 The height is x.

00:26 And we're told to find the volume of this box.

00:29 And since it's a square box, we know that if this side is 24 minus 2x, this side is also going to be 24 minus 2x.

00:39 So to find the volume, it's going to be the.

00:44 Area of the square times height, which is the area is going to be 24 minus 2x square and the height is x.

00:57 So therefore, the volume is equals to x into 24 minus 2x square.

01:07 Now, before this shape became a box, we're told that it was simply a square with side 24.

01:15 That was folded.

01:19 So it was folded into about half to make this box.

01:25 So we know that the height x is going to be less than 12, which is half of the length of the side before it was folded.

01:38 We know that that's going to be less than 12, but it's greater than 0.

01:46 Now the next thing we have to do is plot a graph.

01:54 To plot this graph, we have to substitute different values of x to find the different values of the volume so we know that um v is equals to x into 24 minus 2 x so if x is 1 v is going to be equals to 1 into 24 minus 2 which is going to be equals to square which is equals to 484 if x is equals to 2 v is equals to 2 into 24 minus 2 times 2 is going to be 4 square which is equals to 800 if x is equals to 3 v is equals to 2 into 24 3 into 24 3 into 24 minus 3 times 2 is going to give us 6 square, which is equals to 9, 7, 2.

03:09 If x is equals to 4, v is equal to 4 into 24 minus 8 square, which is equals to 10 to 4.

03:26 Let's keep going if x is equals to five v is equals to five into twenty four minus ten square which is equals to nine eighty if x is equals to six v is equals to six into twenty four minus twelve square which is equals to eight six four so now we've gotten sufficient point for x and y so we can now plot our graph this is going to be our x and v is going to be our y values so we have a graph here x y i'm just going to fill in different values for both axes and then we can plot so i've gone ahead and filled in different values um for different exist for the x and the y exist now we're going to plus our graph when x is 1 y is 484 so when x is 1 when x is 1 y is 484 so it's going to be somewhere here that's our first point i'm just going to mark you with x when x is 2 y is 800 so when x is 2 y is 800 so why is 800 so we have this point right here.

05:18 When y is when x is 3, y is 972...

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