/*! 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 22 A rectangular page is to contain... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 22

A rectangular page is to contain 30 square inches of print. The margins at the top and bottom of the page are to be 2 inches wide. The margins on each side are to be 1 inch wide. Find the dimensions of the page such that the least amount of paper is used.

Short Answer

Expert verified
The page's dimensions that would minimize the use of paper are the width found in step 3 and the length found in step 5.

Step by step solution

01

Define Variables

Denote the width of the page as \( w \), and the length of the page as \( l \). The print area is \( (w-2) \times (l-4) = 30 \) square inches, which gives expression of length using the width, \( l = 4 + \frac{30}{w-2} \).
02

Obtain Area

As the objective is to minimize the area of the page, formulate the area of the page function \( A(w) = w \times l \) using the expression found for the length. Therefore, \( A(w) = w \times (4 + \frac{30}{ w-2}) = 4w + \frac{30w}{w-2} \).
03

Minimize Area

To minimize the area, differentiate the area function w.r.t width and equate it to zero to find the critical points. The derivative \( A'(w) = 4 + \frac{30}{w-2} - \frac{60}{(w-2)^2} \). To solve \( A'(w) = 0 \) for \( w \) yields the width of the page.
04

Test Critical Points

Substitute derivative's critical points and the endpoints ( \( w = 2 \) will be the smallest possible width allowed due to margins) into second derivative \( A''(w)=\frac{120}{(w-2)^3} \) to determine the minimum. A positive value means the function is concave up there and we have found a minimum.
05

Find Length

Now that we know the width that minimizes the area, substitute \( w \) into \( l = 4 + \frac{30}{ w-2} \) to find the corresponding length.

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

The cost and revenue functions for a product are \(C=25.5 x+1000\) and \(R=75.5 x\) (a) Find the average profit function \(\bar{P}=\frac{R-C}{x}\). (b) Find the average profits when \(x\) is 100,500 , and 1000 . (c) What is the limit of the average profit function as \(x\) approaches infinity? Explain your reasoning.

The table lists the average monthly Social Security benefits \(B\) (in dollars) for retired workers aged 62 and over from 1998 through 2005 . A model for the data is \(B=\frac{582.6+38.38 t}{1+0.025 t-0.0009 t^{2}}, \quad 8 \leq t \leq 15\) where \(t=8\) corresponds to 1998 . $$ \begin{array}{|l|l|l|l|l|l|l|l|l|} \hline t & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 \\ \hline B & 780 & 804 & 844 & 874 & 895 & 922 & 955 & 1002 \\ \hline \end{array} $$ (a) Use a graphing utility to create a scatter plot of the data and graph the model in the same viewing window. How well does the model fit the data? (b) Use the model to predict the average monthly benefit in \(2008 .\) (c) Should this model be used to predict the average monthly Social Security benefits in future years? Why or why not?

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. \(y=x^{5 / 3}-5 x^{2 / 3}\)

The cost function for a company to recycle \(x\) tons of material is given by \(C=1.25 x+10,500\), where \(C\) is measured in dollars. (a) Find the average cost function \(\bar{C}\). (b) Find the average costs of recycling 100 tons of material and 1000 tons of material. (c) Determine the limit of the average cost function as \(x\) approaches infinity. Interpret the limit in the context of the problem.

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

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation

Pure Maths

Read Explanation

Statistics

Read Explanation

Probability and Statistics

Read Explanation

Geometry

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.