\( \newcommand{\abs}[1]{\left| \, {#1} \, \right| } \) \( \newcommand{\cm}{\mathrm{cm} } \) \( \newcommand{\sec}{ \, \mathrm{sec} \, } \) \( \newcommand{\units}[1]{\,\text{#1}} \) \( \newcommand{\vhat}[1]{\,\hat{#1}} \) \( \newcommand{\vhati}{\,\hat{i}} \) \( \newcommand{\vhatj}{\,\hat{j}} \) \( \newcommand{\vhatk}{\,\hat{k}} \) \( \newcommand{\vect}[1]{\boldsymbol{\vec{#1}}} \) \( \newcommand{\norm}[1]{\|{#1}\|} \) \( \newcommand{\arccot}{ \, \mathrm{arccot} \, } \) \( \newcommand{\arcsec}{ \, \mathrm{arcsec} \, } \) \( \newcommand{\arccsc}{ \, \mathrm{arccsc} \, } \) \( \newcommand{\sech}{ \, \mathrm{sech} \, } \) \( \newcommand{\csch}{ \, \mathrm{csch} \, } \) \( \newcommand{\arcsinh}{ \, \mathrm{arcsinh} \, } \) \( \newcommand{\arccosh}{ \, \mathrm{arccosh} \, } \) \( \newcommand{\arctanh}{ \, \mathrm{arctanh} \, } \) \( \newcommand{\arccoth}{ \, \mathrm{arccoth} \, } \) \( \newcommand{\arcsech}{ \, \mathrm{arcsech} \, } \) \( \newcommand{\arccsch}{ \, \mathrm{arccsch} \, } \)

17Calculus Derivatives - Optimizing Areas

17Calculus
Single Variable Calculus
Derivatives
Integrals
Multi-Variable Calculus
Precalculus
Functions

Before going through this page, make sure you have thoroughly read and understand the basics of optimization. The practice problems on that page will also give you a head start on these types of problems.

Getting Started

The idea with optimization problems is to get one equation with two variables. One variable should be the value you are trying to maximize (or minimize) and the other variable can be anything but there is often more than one way to set up the equation. Then, you take the derivative with respect to the second variable, set it equal to zero and solve.

Before we get started, this first panel will help you review word problems by giving some general advice and techniques. Even if you already know how to work word problems, you may find the information in this panel helpful.

Guidelines For Working Word Problems

Word problems are what math students dread the most. I completely understand. I had a lot of difficulty too. However, I found a technique so that I was not only able to successfully work word problems but, eventually, I came to like them and am now able to teach them.

I will tell you up front that figuring out how to work word problems is not easy and takes some independent work on your part to master them. But once you do, you will find them enjoyable and, since so many students struggle with them, most teachers give pretty easy problems, even on exams. So you should be able to breeze your way through them.

First, what doesn't work. Most books try to lump all word problems together and give you general guidelines on how to work them. I have NEVER found that helpful. It wasn't until I was able separate out the different types of word problems, that I came to understand how to work them. Since there are different types of word problems, there are different ways to work them.

Here is what you need to do.

1. Find plenty of problems with worked out solutions. Here are some suggested resources.
- Get a good book with examples and worked out solutions of the type of word problems you are studying. We have posted several suggestions on the books page.
- Check out the solution manual for problems in your textbook.
- 17calculus practice problems

2. Once you have a good selection of worked out solutions, go through them carefully and pick up patterns on how they set up the problems, solve them and give the final answer. Pick the ones that are similar to ones in your textbook that you are working on for your class.

3. Key - - Group the problems into categories that make sense to you. Some examples might be problems with triangles, problems with right circular cylinders, problems asking you to find areas or volumes. A single problem can go into multiple categories based on configuration or type of question or any other category that makes sense to you.

4. Work the problems yourself before looking at the solutions. Then compare your solutions with the book. Determine what you did wrong and what you need to learn in order to work the problems correctly. At first, this will be slow and painful but once your brain catches on, it will start to be fun. Be patient with yourself, work hard and don't give up. [ In the case of videos, stop the video after the presenter has given the problem statement and work it yourself before watching them solve it. ]

5. Important - - Once you have finished a problem, write down the meaning of your answer in words and then reread the problem statement to make sure that your answer is what the problem asked for, including units. For example, I worked a problem about a skydiver and the problem asked for the time it takes for the skydiver to hit the ground after he opens his parachute. When I finished the problem, I had calculated the time that it takes for the skydiver to hit the ground since he jumped out of the plane. When I checked my answer in the back of the book, I was confused until I realized that the number I had was not what the problem asked for. If I had written what my answer means in words and then looked back at the problem, I would have realized right away what I needed to do to finish the problem. Doing this will save you from losing points on homework and exams and it takes only a few seconds.

6. Make sure you understand every single step and, when looking at the solution, figure out why they do things the way they do. If you made a mistake, try to understand what your mistake was and what you need to understand in order to not make the same mistake again. [ Also remember that no textbook or video is always 100% correct. If you can not figure out your mistake, find someone to ask and see if the solution manual is incorrect. ]

7. Pick up patterns and general ideas from each group of problems by working the same type of problems all together. Don't jump around to different types. Stay with one type for several problems. I won't tell you exactly how many. You need to determine that by how difficult the problems are, how well you think you understand the current type, how much time you have and how well you want to do on your homework and exams. Sometimes you can go on after working 5 of the same type, sometimes it takes 10 or more.

8. Find a friend to work with and go over the problems with them AFTER you have worked them on your own. Remember, at exam time you will be on your own. So don't rely on someone else too much. If you know more than the other person, explain your work to them. Communicating your work to someone else helps you understand it better. If you know less, ask lots of questions and ask them to explain their solution to you.

9. Do NOT do shortcuts. Shortcuts are good AFTER you have learned the material, not while you are learning the material. Do it the long way for a while until you are know it really well.

10. 2nd Key - - Do not just look at the solutions or watch someone else work the problems. You need to get out a pencil and paper and work them yourself. You are going to get frustrated. You are going to want to quit, but don't quit. Use that feeling to motivate yourself and show yourself that you can do it. It feels great to master something that is difficult. If you have never pushed through something difficult before, try it now. It is not easy but it is worth it. I know because I went through this same process myself.

11. Finally, do not skip ANYTHING and NEVER GIVE UP. Make sure you understand every single step in every single problem. Here's why: Chances are, if you skip something, it will show up on an exam precisely because the part you don't understand is probably the most difficult part of the problem and teachers expect you to skip it. So they put it on exams to see if you understand the difficult parts.

So far, I have found that implementing these ideas as the best way to figure out how to work word problems. There are tons of general guidelines in books (most likely in your textbook too) that never really helped me. Give this technique a try. Remember, you are now in charge of your own learning. No one is going to help you from here on out. You need to do it.

Optimizing Areas

Optimizing area is one of the most common types of problems. Many times they involve rectangles, triangles or other common geometric figures.

Here is a video with an example of maximizing area. This is interesting because it involves both rectangles and triangles.

PatrickJMT - Optimization Problem #3 - Making a Rain Gutter [10min-21secs]

video by PatrickJMT

Here is a table listing some of the most common area equations you will need to solve this type of problem.

Rectangles

\(A=lw\)

Triangles

\(A=bh/2\)

Circles

\(A=\pi r^2 \)

Surface Area of a Sphere

\( A = 4\pi r^2 \)

Parallelogram

\( A = bh \)

More Area Equations

Wikipedia - Area

From the area of a rectangle, you should know the area of a square, right?
Okay, time for some practice problems maximizing area.

The Practicing Mind: Developing Focus and Discipline in Your Life - Master Any Skill or Challenge by Learning to Love the Process

Practice

Basic

Calculate the area of the largest rectangle that can be inscribed inside the ellipse \(\displaystyle{\frac{x^2}{4}+\frac{y^2}{9}=1}\).

Problem Statement

Calculate the area of the largest rectangle that can be inscribed inside the ellipse \(\displaystyle{\frac{x^2}{4}+\frac{y^2}{9}=1}\).

Final Answer

Area = 12 square units

Problem Statement

Calculate the area of the largest rectangle that can be inscribed inside the ellipse \(\displaystyle{\frac{x^2}{4}+\frac{y^2}{9}=1}\).

Solution

PatrickJMT - 2098 video solution

video by PatrickJMT

Final Answer

Area = 12 square units

Log in to rate this practice problem and to see it's current rating.

Find the dimensions of a rectangle with perimeter 100m where the area is as large as possible.

Problem Statement

Find the dimensions of a rectangle with perimeter 100m where the area is as large as possible.

Solution

Krista King Math - 1121 video solution

video by Krista King Math

Log in to rate this practice problem and to see it's current rating.

The area of a rectangle is given by \( A(x)=16x-x^2 \) where \(x\) is the length of one of the sides. What is the maximum area of this rectangle?

Problem Statement

The area of a rectangle is given by \( A(x)=16x-x^2 \) where \(x\) is the length of one of the sides. What is the maximum area of this rectangle?

Final Answer

64 square units

Problem Statement

The area of a rectangle is given by \( A(x)=16x-x^2 \) where \(x\) is the length of one of the sides. What is the maximum area of this rectangle?

Solution

PatrickJMT - 2100 video solution

video by PatrickJMT

Final Answer

64 square units

Log in to rate this practice problem and to see it's current rating.

A 20 inch wire is cut into two (not necessarily equal) pieces and shaped into two squares. What is the minimum possible sum of the two areas?

Problem Statement

A 20 inch wire is cut into two (not necessarily equal) pieces and shaped into two squares. What is the minimum possible sum of the two areas?

Solution

PatrickJMT - 1122 video solution

video by PatrickJMT

Log in to rate this practice problem and to see it's current rating.

Suppose a farmer wants to build a rectangular pen for his cows with 500ft of fencing. If one side of the pen is along a river (no fencing required), what is the area of the largest pen he can build?

Problem Statement

Suppose a farmer wants to build a rectangular pen for his cows with 500ft of fencing. If one side of the pen is along a river (no fencing required), what is the area of the largest pen he can build?

Solution

PatrickJMT - 1125 video solution

video by PatrickJMT

Log in to rate this practice problem and to see it's current rating.

A farmer with 750ft of fencing wants to enclose a rectangular area and then divide it into four pens. What is the largest total area of the four pens?

Problem Statement

A farmer with 750ft of fencing wants to enclose a rectangular area and then divide it into four pens. What is the largest total area of the four pens?

Solution

The Organic Chemistry Tutor - 3875 video solution

Log in to rate this practice problem and to see it's current rating.

A piece of wire 10m long is cut into two pieces. One piece is bent into a square and the other is bent into an equilateral triangle. How should the wire be cut so that the total area is a maximum? A minimum?

Problem Statement

A piece of wire 10m long is cut into two pieces. One piece is bent into a square and the other is bent into an equilateral triangle. How should the wire be cut so that the total area is a maximum? A minimum?

Solution

This is quite an interesting problem. She is calculating the maximum and minimum of the combination of two figures, not just one. However, the added complexity does not really make the problem as difficult as it might initially seem.

Krista King Math - 1135 video solution

video by Krista King Math

Log in to rate this practice problem and to see it's current rating.

A cylindrical can must be made to hold 1L of oil. Find the dimensions that minimize the cost of the metal used to make it.

Problem Statement

A cylindrical can must be made to hold 1L of oil. Find the dimensions that minimize the cost of the metal used to make it.

Solution

Krista King Math - 1136 video solution

video by Krista King Math

Log in to rate this practice problem and to see it's current rating.

Find the dimensions that will minimize the cost of metal to make a cylindrical can that is able to store 0.25L of fluid.

Problem Statement

Find the dimensions that will minimize the cost of metal to make a cylindrical can that is able to store 0.25L of fluid.

Solution

The Organic Chemistry Tutor - 3877 video solution

Log in to rate this practice problem and to see it's current rating.

Intermediate

Calculate the dimensions of an open-topped rectangular box that minimizes the surface area. The volume is 972in3 and the length of the bottom is twice as long as the width.

Problem Statement

Calculate the dimensions of an open-topped rectangular box that minimizes the surface area. The volume is 972in3 and the length of the bottom is twice as long as the width.

Solution

Krista King Math - 1131 video solution

video by Krista King Math

Log in to rate this practice problem and to see it's current rating.

Find the area of the largest rectangle with its base on the x-axis and the other two vertices above the x-axis on the parabola \(y=8-x^2\).

Problem Statement

Find the area of the largest rectangle with its base on the x-axis and the other two vertices above the x-axis on the parabola \(y=8-x^2\).

Solution

PatrickJMT - 1124 video solution

video by PatrickJMT

Log in to rate this practice problem and to see it's current rating.

Find the area of the largest rectangle that can be inscribed in a semicircle of radius r.

Problem Statement

Find the area of the largest rectangle that can be inscribed in a semicircle of radius r.

Solution

Krista King Math - 1133 video solution

video by Krista King Math

Log in to rate this practice problem and to see it's current rating.

A book publisher wants each page of a book to have margins of 1 inch at the top and along each side, and a 1.5 inch margin at the bottom. The printed area inside the margins is to be 80 in2. What dimensions will minimize the size of each page?

Problem Statement

A book publisher wants each page of a book to have margins of 1 inch at the top and along each side, and a 1.5 inch margin at the bottom. The printed area inside the margins is to be 80 in2. What dimensions will minimize the size of each page?

Solution

Krista King Math - 1132 video solution

video by Krista King Math

Log in to rate this practice problem and to see it's current rating.

Find the dimensions of the rectangle with the largest area that can be inscribed in an equilateral triangle of side L, if one side of the rectangle lies on the base of the triangle.

Problem Statement

Find the dimensions of the rectangle with the largest area that can be inscribed in an equilateral triangle of side L, if one side of the rectangle lies on the base of the triangle.

Solution

Krista King Math - 1134 video solution

video by Krista King Math

Log in to rate this practice problem and to see it's current rating.

An open top box with volume 20ft3 is twice as long as it is wide. Find the width that minimizes the amount of material required to make the box.

Problem Statement

An open top box with volume 20ft3 is twice as long as it is wide. Find the width that minimizes the amount of material required to make the box.

Final Answer

\(\sqrt[3]{15} \approx 2.466\) ft

Problem Statement

An open top box with volume 20ft3 is twice as long as it is wide. Find the width that minimizes the amount of material required to make the box.

Solution

Krista King Math - 2110 video solution

video by Krista King Math

Final Answer

\(\sqrt[3]{15} \approx 2.466\) ft

Log in to rate this practice problem and to see it's current rating.

A norman window has the shape of a semi-circle on top of a rectangle. Find the dimensions of the window that will maximize the area if the perimeter is 20ft.

Problem Statement

A norman window has the shape of a semi-circle on top of a rectangle. Find the dimensions of the window that will maximize the area if the perimeter is 20ft.

Solution

The Organic Chemistry Tutor - 3881 video solution

Log in to rate this practice problem and to see it's current rating.

Really UNDERSTAND Calculus

Log in to rate this page and to see it's current rating.

Topics You Need To Understand For This Page

Related Topics and Links

related topics on other pages

related rates

external links you may find helpful

optimization youtube playlist

WikiBooks: Optimization

To bookmark this page and practice problems, log in to your account or set up a free account.

Search Practice Problems

Do you have a practice problem number but do not know on which page it is found? If so, enter the number below and click 'page' to go to the page on which it is found or click 'practice' to be taken to the practice problem.

more calculus help

Kindle Unlimited Membership Plans

As an Amazon Associate I earn from qualifying purchases.

I recently started a Patreon account to help defray the expenses associated with this site. To keep this site free, please consider supporting me.

Support 17Calculus on Patreon

Practice Search

Do NOT follow this link or you will be banned from the site!

When using the material on this site, check with your instructor to see what they require. Their requirements come first, so make sure your notation and work follow their specifications.

DISCLAIMER - 17Calculus owners and contributors are not responsible for how the material, videos, practice problems, exams, links or anything on this site are used or how they affect the grades or projects of any individual or organization. We have worked, to the best of our ability, to ensure accurate and correct information on each page and solutions to practice problems and exams. However, we do not guarantee 100% accuracy. It is each individual's responsibility to verify correctness and to determine what different instructors and organizations expect. How each person chooses to use the material on this site is up to that person as well as the responsibility for how it impacts grades, projects and understanding of calculus, math or any other subject. In short, use this site wisely by questioning and verifying everything. If you see something that is incorrect, contact us right away so that we can correct it.

Links and banners on this page are affiliate links. We carefully choose only the affiliates that we think will help you learn. Clicking on them and making purchases help you support 17Calculus at no extra charge to you. However, only you can decide what will actually help you learn. So think carefully about what you need and purchase only what you think will help you.

We use cookies on this site to enhance your learning experience.

17calculus

Copyright © 2010-2022 17Calculus, All Rights Reserved     [Privacy Policy]     [Support]     [About]

mathjax.org
Real Time Web Analytics