Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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2.1 10
![### Educational Website Transcription
#### Table Data
| x | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| V | 484 | 800 | 972 | 1,024 | 980 | 864 |
#### Tasks
**(a)** Write the volume (V) in cubic centimeters.
**(b)** Plot the points (x, V) from the table in part (a). Does the relation defined by the ordered pairs represent a function of x?
- Yes
- No
If it does represent a function of x, write the function. (If an answer does not exist, enter DNE.)
\[ V = \]
Determine the domain of the function.
- V is not a function of x.
- All real numbers.
- All real numbers except \( x = 12 \).
- \( 0 < x \le 12 \).
- All real numbers except \( x = 0 \).
#### Graphs and Diagrams Description
There are no specific graphs or diagrams provided in the original image; however, students are encouraged to plot the given table values onto a graph, with x-values on the horizontal axis and V-values on the vertical axis. The goal is to visually determine whether the plotted points suggest a functional relationship between x and V.](https://content.bartleby.com/qna-images/question/3aa1f949-9b51-4eeb-bd19-8a93150eb874/27e8f24e-8e3c-46b6-99b9-f32a0b688a3e/rdiqex_processed.jpeg)
Transcribed Image Text:### Educational Website Transcription
#### Table Data
| x | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| V | 484 | 800 | 972 | 1,024 | 980 | 864 |
#### Tasks
**(a)** Write the volume (V) in cubic centimeters.
**(b)** Plot the points (x, V) from the table in part (a). Does the relation defined by the ordered pairs represent a function of x?
- Yes
- No
If it does represent a function of x, write the function. (If an answer does not exist, enter DNE.)
\[ V = \]
Determine the domain of the function.
- V is not a function of x.
- All real numbers.
- All real numbers except \( x = 12 \).
- \( 0 < x \le 12 \).
- All real numbers except \( x = 0 \).
#### Graphs and Diagrams Description
There are no specific graphs or diagrams provided in the original image; however, students are encouraged to plot the given table values onto a graph, with x-values on the horizontal axis and V-values on the vertical axis. The goal is to visually determine whether the plotted points suggest a functional relationship between x and V.
![### Open Box Maximum Volume Calculation
An open box of maximum volume is made from a square piece of material 24 centimeters on a side by cutting equal squares from the corners and turning up the sides.
#### Diagram Description:
- The left diagram represents a square piece of material with side length 24 cm. Squares of side length \( x \) cm are cut out from each corner.
- The right diagram shows the resulting box when the sides are turned up. The dimensions of the box are:
- Height: \( x \) cm
- Length and Width: \( 24 - 2x \) cm, as the squares of side length \( x \) cm are removed from each corner.
#### (a) Volume Table for Various Heights
The table below shows the volumes \( V \) (in cubic centimeters) of the box for various heights \( x \) (in centimeters). Use the table to estimate the maximum volume.
| **Height, \( x \) (cm)** | 1 | 2 | 3 | 4 | 5 | 6 |
|--------------------------|-----|-----|-----|------|------|------|
| **Volume, \( V \) (cm³)**| 484 | 800 | 972 | 1,024| 980 | 864 |
**Estimated Maximum Volume:**
\[ \boxed{1,024} \] cubic centimeters
#### (b) Plot Points and Determine Function
- **Plot the points \((x, V)\) from the table in part (a).**
- **Question: Does the relation defined by the ordered pairs represent \( V \) as a function of \( x \)?**
- Options:
- Yes
- No
- **If it does represent a function of \( x \), write the function. (If an answer does-not exist, enter DNE.)**
\[ V = \]
(Note: The function representing \( V \) in terms of \( x \) can typically be derived using algebraic methods, involving terms representing the volume of the box, where \( V = x(24-2x)^2 \). The maximum volume can usually be found by taking the derivative and setting it to zero to solve for \( x \).)](https://content.bartleby.com/qna-images/question/3aa1f949-9b51-4eeb-bd19-8a93150eb874/27e8f24e-8e3c-46b6-99b9-f32a0b688a3e/ihzcmx_processed.jpeg)
Transcribed Image Text:### Open Box Maximum Volume Calculation
An open box of maximum volume is made from a square piece of material 24 centimeters on a side by cutting equal squares from the corners and turning up the sides.
#### Diagram Description:
- The left diagram represents a square piece of material with side length 24 cm. Squares of side length \( x \) cm are cut out from each corner.
- The right diagram shows the resulting box when the sides are turned up. The dimensions of the box are:
- Height: \( x \) cm
- Length and Width: \( 24 - 2x \) cm, as the squares of side length \( x \) cm are removed from each corner.
#### (a) Volume Table for Various Heights
The table below shows the volumes \( V \) (in cubic centimeters) of the box for various heights \( x \) (in centimeters). Use the table to estimate the maximum volume.
| **Height, \( x \) (cm)** | 1 | 2 | 3 | 4 | 5 | 6 |
|--------------------------|-----|-----|-----|------|------|------|
| **Volume, \( V \) (cm³)**| 484 | 800 | 972 | 1,024| 980 | 864 |
**Estimated Maximum Volume:**
\[ \boxed{1,024} \] cubic centimeters
#### (b) Plot Points and Determine Function
- **Plot the points \((x, V)\) from the table in part (a).**
- **Question: Does the relation defined by the ordered pairs represent \( V \) as a function of \( x \)?**
- Options:
- Yes
- No
- **If it does represent a function of \( x \), write the function. (If an answer does-not exist, enter DNE.)**
\[ V = \]
(Note: The function representing \( V \) in terms of \( x \) can typically be derived using algebraic methods, involving terms representing the volume of the box, where \( V = x(24-2x)^2 \). The maximum volume can usually be found by taking the derivative and setting it to zero to solve for \( x \).)
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