(a) Draw several diagrams to illustrate the situation, some short boxes with large bases and some tall boxes with small bases. Find the volume of each configuration. Does it appear that there is a maximum volume? If so, estimate it. (b) Draw a diagram illustrating the general situation. Let x denote the length of the side of the square being cut out. Let y denote the length of the base. (c) Write an expression for the volume V in terms of both x and y. V = (d) Use the given information to write an equation that relates the variables x and y. (e) Use part (d) to write the volume as a function of only x. V(x) - (f) Finish solving the problem by finding the largest volume (in ft) that such a box can have. V=
(a) Draw several diagrams to illustrate the situation, some short boxes with large bases and some tall boxes with small bases. Find the volume of each configuration. Does it appear that there is a maximum volume? If so, estimate it. (b) Draw a diagram illustrating the general situation. Let x denote the length of the side of the square being cut out. Let y denote the length of the base. (c) Write an expression for the volume V in terms of both x and y. V = (d) Use the given information to write an equation that relates the variables x and y. (e) Use part (d) to write the volume as a function of only x. V(x) - (f) Finish solving the problem by finding the largest volume (in ft) that such a box can have. 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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![### Box Construction Problem
Consider the following problem: a box with an open top is to be constructed from a square piece of cardboard, 3 feet wide, by cutting out a square from each of the four corners and bending up the sides. Find the largest volume that such a box can have.
**(a)** Draw several diagrams to illustrate the situation, some short boxes with large bases and some tall boxes with small bases. Find the volume of each configuration. Does it appear that there is a maximum volume? If so, estimate it.
**(b)** Draw a diagram illustrating the general situation. Let \( x \) denote the length of the side of the square being cut out. Let \( y \) denote the length of the base.
**(c)** Write an expression for the volume \( V \) in terms of both \( x \) and \( y \).
\[ V = \text{______} \]
**(d)** Use the given information to write an equation that relates the variables \( x \) and \( y \).
\[ \text{______} \]
**(e)** Use part (d) to write the volume as a function of only \( x \).
\[ V(x) = \text{______} \]
**(f)** Finish solving the problem by finding the largest volume (in ft\(^3\)) that such a box can have.
\[ V = \text{______} \, \text{ft}^3 \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb97d67e1-dda9-48ae-b6f6-b8910f9a7c16%2F2e00134c-8642-458c-a80e-030e4dcc39dc%2Fg3kq2wc_processed.png&w=3840&q=75)
Transcribed Image Text:### Box Construction Problem
Consider the following problem: a box with an open top is to be constructed from a square piece of cardboard, 3 feet wide, by cutting out a square from each of the four corners and bending up the sides. Find the largest volume that such a box can have.
**(a)** Draw several diagrams to illustrate the situation, some short boxes with large bases and some tall boxes with small bases. Find the volume of each configuration. Does it appear that there is a maximum volume? If so, estimate it.
**(b)** Draw a diagram illustrating the general situation. Let \( x \) denote the length of the side of the square being cut out. Let \( y \) denote the length of the base.
**(c)** Write an expression for the volume \( V \) in terms of both \( x \) and \( y \).
\[ V = \text{______} \]
**(d)** Use the given information to write an equation that relates the variables \( x \) and \( y \).
\[ \text{______} \]
**(e)** Use part (d) to write the volume as a function of only \( x \).
\[ V(x) = \text{______} \]
**(f)** Finish solving the problem by finding the largest volume (in ft\(^3\)) that such a box can have.
\[ V = \text{______} \, \text{ft}^3 \]
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