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- NoIf 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 DescriptionThere 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 CalculationAn 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 HeightsThe 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 \( x \).)

The volume of a cuboid=length×width×height

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 side 24-2x. x 24-2x x (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. Height, x 1 2 3 4 5 6 Volume, V 484 800 972 1,024 980 864 cubic centimeters (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? Yes O No If it does represent a function of x, write the function. (If an answer does not exist, enter DNE.) V =

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 side 24-2x. x 24-2x x (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. Height, x 1 2 3 4 5 6 Volume, V 484 800 972 1,024 980 864 cubic centimeters (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? Yes O No If it does represent a function of x, write the function. (If an answer does not exist, enter DNE.) V =

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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