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

17Calculus Derivatives - Optimizing Volumes

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

These types of problems are pretty obvious. You have an equation with a volume that you need to maximize. It might help for you to have a sheet of equations of volumes that come up as you work these problems like cones, cylinders and spheres. Here are some equations you may need.

Rectangular Container

\( V = lwh \)

Sphere

\(\displaystyle{ V = \frac{4}{3} \pi r^3 }\)

Circular Cylinder

\( V = \pi r^2 h \)

Cone

\(\displaystyle{ V = \frac{1}{3}\pi r^2 h }\)

Practice

Basic

A woman is building a box with an open top by removing squares of equal size from the corners of a sheet of metal and folding the sides upwards. If the sheet of metal is 2ft by 2ft, what is the maximum possible volume of such a box?

Problem Statement

A woman is building a box with an open top by removing squares of equal size from the corners of a sheet of metal and folding the sides upwards. If the sheet of metal is 2ft by 2ft, what is the maximum possible volume of such a box?

Solution

PatrickJMT - 1126 video solution

video by PatrickJMT

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Suppose a can is made up of \(25.625\pi\) square inches of material. What dimensions would maximize the volume of this cylindrical can?

Problem Statement

Suppose a can is made up of \(25.625\pi\) square inches of material. What dimensions would maximize the volume of this cylindrical can?

Solution

PatrickJMT - 1127 video solution

video by PatrickJMT

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The total of the length and girth of a box cannot exceed 120in. What dimensions maximize volume if the end is square?

Problem Statement

The total of the length and girth of a box cannot exceed 120in. What dimensions maximize volume if the end is square?

Final Answer

40in x 20in x 20in

Problem Statement

The total of the length and girth of a box cannot exceed 120in. What dimensions maximize volume if the end is square?

Solution

Krista King Math - 2066 video solution

video by Krista King Math

Final Answer

40in x 20in x 20in

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What is the largest possible volume of a box with a square base and an open top that can be produced using 900cm2 of material?

Problem Statement

What is the largest possible volume of a box with a square base and an open top that can be produced using 900cm2 of material?

Solution

The Organic Chemistry Tutor - 3876 video solution

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A box with an open top is to be constructed from a square piece of cardboard that is 4 feet wide by cutting out a square from each side of the four corners and bending up the sides. Find the maximum volume of the box.

Problem Statement

A box with an open top is to be constructed from a square piece of cardboard that is 4 feet wide by cutting out a square from each side of the four corners and bending up the sides. Find the maximum volume of the box.

Solution

The Organic Chemistry Tutor - 3880 video solution

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Find the volume of the largest open top box that can be made from a piece of cardboard 24 inches square by cutting equal squares from the corners and turning up the sides.

Problem Statement

Find the volume of the largest open top box that can be made from a piece of cardboard 24 inches square by cutting equal squares from the corners and turning up the sides.

Solution

Khan Academy - 1140 video solution

video by Khan Academy

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Intermediate

Find the largest possible volume of a right circular cylinder that can be inscribed in a sphere with radius \(r\).

Problem Statement

Find the largest possible volume of a right circular cylinder that can be inscribed in a sphere with radius \(r\).

Solution

Krista King Math - 1137 video solution

video by Krista King Math

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Find the maximum volume of a cone-shaped drinking cup made from a circular piece of paper of radius \(R\), where a sector has been removed and the outer edges are joined.

Problem Statement

Find the maximum volume of a cone-shaped drinking cup made from a circular piece of paper of radius \(R\), where a sector has been removed and the outer edges are joined.

Solution

Krista King Math - 1138 video solution

video by Krista King Math

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A rectangular sheet of paper with perimeter 39cm is rolled into a cylinder. What dimensions of the sheet would maximize the volume of the cylinder?

Problem Statement

A rectangular sheet of paper with perimeter 39cm is rolled into a cylinder. What dimensions of the sheet would maximize the volume of the cylinder?

Final Answer

13cm x 6.5cm

Problem Statement

A rectangular sheet of paper with perimeter 39cm is rolled into a cylinder. What dimensions of the sheet would maximize the volume of the cylinder?

Solution

Krista King Math - 2078 video solution

video by Krista King Math

Final Answer

13cm x 6.5cm

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Really UNDERSTAND Calculus

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Topics You Need To Understand For This Page

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