/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 19 An open box is to be made from a... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 19

An open box is to be made from a two-foot by three-foot rectangular piece of material by cutting equal squares from the corners and turning up the sides. Find the volume of the largest box that can be made in this manner.

Short Answer

Expert verified
The volume of the largest box that can be constructed from the given rectangular material is calculated by following the steps mentioned above, and it will be obtained after substituting the optimal \( x \) value into the volume function \( V(x) \). The exact value depends on the solution of the derivative equations.

Step by step solution

01

Setting Up the Function for Volume

To set up the volume function, let's denote the length of the cut square sides as \( x \). After the cuts, the box's dimensions will be: length \(3 - 2x\), width \(2 - 2x\) and height \(x\). So the volume function, \( V \), is: \( V(x) = x \cdot (3 - 2x) \cdot (2 - 2x) \).
02

Calculating the Derivative

Our goal is to maximize \( V \), for which we will have to calculate the first derivative of the \( V(x) \) function. This will give us critical points that may be the maxima or minima of the function. The derivative, \( V'(x) \), can be found by applying the product rule.
03

Finding Critical Points

For the function to be a maximum, \( V'(x) \) must be equal to 0. Thus, solve \( V'(x) = 0 \), to find the values of \( x \) at critical points.
04

Verification of Maximum Point

We need to verify if the critical point found in Step 3 gives the maximum volume. Do this by using the second derivative test, which describes that if \( V''(x) < 0 \), then \( V(x) \) is a maximum.
05

Compute Maximum Volume

Substitute the value of \( x \) which gives the maximum volume into the volume function, \( V(x) \), to compute the largest possible volume of the box.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Use a graphing utility to graph the function. Choose a window that allows all relative extrema and points of inflection to be identified on the graph. \(y=x^{4 / 3}\)

Use a graphing utility to graph the function. Choose a window that allows all relative extrema and points of inflection to be identified on the graph. \(y=\frac{x^{2}}{x^{2}+3}\)

Use a graphing utility to graph the function. Choose a window that allows all relative extrema and points of inflection to be identified on the graph. \(y=x^{5 / 3}-5 x^{2 / 3}\)

Find the differential \(d y\). \(y=\sqrt{9-x^{2}}\)

Sketch the graph of the function. Choose a scale that allows all relative extrema and points of inflection to be identified on the graph. \(y=-x^{2}-2 x+3\)
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Calculus

Read Explanation

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Pure Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.