I have good news for you. You already know how to solve optimization problems if you have worked through graphing and understand how to use the derivative to find minimums and maximums. The idea is the same. Maximums and minimums occur at points with the derivative is zero, called critical points. So, the key to these problems is just setting them up. Once set up there are a few things you can do to make the problem easier to solve.
If you want a complete lecture on optimization, we recommend this video from one of our favorite instructors.
video by Prof Leonard |
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Recommended Books on Amazon (affiliate links) | ||
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Why Learn Optimization?
Most of the time in college you are told to learn something without any real reason for learning it. This is normal and, most of the time, necessary since 'when will I ever use this?' is often difficult or impossible to answer. However, when it comes to optimization, there are some direct applications that you can understand now. Here is a video explaining some of them. This may not satisfy some of you but for many of you, engineers especially, it is nice to see a little bit of the path ahead to make your time feel better spent. Enjoy!
video by Zach Star |
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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.
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.
So far these types of problems sound a lot like related rates. But let's think about it. The word 'rate' in related rates implies that something is moving and we are trying to related the rates of two (or more) things that are moving. However, with optimization, nothing is really moving. What we are doing is setting up the equations for some type of geometry and then adjusting that figure until something optimizes (either minimizes or maximizes). Cool, eh? So let's look at some possible geometries.
Okay, so how do you get started? Well, you already know how to take the derivative of functions. You know enough algebra to be able to solve the resulting equation when it is set to zero. All you need to learn is how to set up the equations. The best way to do that is to watch a few examples and try working some problems on your own. If you have gone through our related rates pages, you know that we suggested categorizing problems by what makes sense to you. We apply that same idea here. We group problems based on the geometry of the situation.
Before we get started with discussion of specific cases, let's watch a video explaining optimization in general and how to work these types of problems. This is a great video that explains things clearly including an example in the second half of the video. We think this video will help you a lot.
video by Krista King Math |
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Optimizing Various Parameters
Problems of this type might optimize things like angle, time, cost and, a very common one, relationship between two numbers. Many books start out the discussion of optimization with discussing this last type since the equations are usually pretty easy to set up.
Here is a video proving Snell's law. You need only calculus that you know so far to prove this law. This video is interesting but is not required for understanding optimization.
video by PatrickJMT |
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Okay, let's work some practice problems. Working these before going on to distances, areas and volumes will give you a head start on those other pages. Once you have worked through these problems, your next logical step is to learn how to optimize distances.
Practice
Basic |
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Find two real numbers with difference 20 and minimum possible product.
Problem Statement
Find two real numbers with difference 20 and minimum possible product.
Solution
video by Krista King Math |
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A farmer needs to build a rectangular enclosure of 750ft^{2}. Fencing costs $5/ft for opposite sides and $8/ft for the other sides. Find the dimensions that minimize the cost.
Problem Statement |
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A farmer needs to build a rectangular enclosure of 750ft^{2}. Fencing costs $5/ft for opposite sides and $8/ft for the other sides. Find the dimensions that minimize the cost.
Final Answer |
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$5/ft side = \(20\sqrt{3}\) ft; $8/ft side = \(25\sqrt{3}/2\) ft
Problem Statement
A farmer needs to build a rectangular enclosure of 750ft^{2}. Fencing costs $5/ft for opposite sides and $8/ft for the other sides. Find the dimensions that minimize the cost.
Solution
video by Krista King Math |
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Final Answer
$5/ft side = \(20\sqrt{3}\) ft; $8/ft side = \(25\sqrt{3}/2\) ft
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A farmer wants to enclose a rectangular area along a highway. The fencing along the highway will made of metal which costs $20/ft. The rest of the area will be fenced with wood which costs $5/ft. What are the dimensions that will minimize the cost to enclose a field with an area of 1600ft^{2}?
Problem Statement
A farmer wants to enclose a rectangular area along a highway. The fencing along the highway will made of metal which costs $20/ft. The rest of the area will be fenced with wood which costs $5/ft. What are the dimensions that will minimize the cost to enclose a field with an area of 1600ft^{2}?
Solution
video by The Organic Chemistry Tutor |
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A company sells \(x\) thousand candy bars at \(p(x)=62-x/12\) cents per bar. How many bars should they sell to maximize revenue, \(R\), where \(R(x)=x[p(x)]\)?
Problem Statement |
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A company sells \(x\) thousand candy bars at \(p(x)=62-x/12\) cents per bar. How many bars should they sell to maximize revenue, \(R\), where \(R(x)=x[p(x)]\)?
Final Answer |
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372 thousand candy bars
Problem Statement
A company sells \(x\) thousand candy bars at \(p(x)=62-x/12\) cents per bar. How many bars should they sell to maximize revenue, \(R\), where \(R(x)=x[p(x)]\)?
Solution
video by Krista King Math |
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Final Answer
372 thousand candy bars
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Find the time when velocity is minimum if velocity is given by \(v(t)=t^2-8t+2\) ft/sec.
Problem Statement |
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Find the time when velocity is minimum if velocity is given by \(v(t)=t^2-8t+2\) ft/sec.
Final Answer |
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\(t=4\) sec
Problem Statement
Find the time when velocity is minimum if velocity is given by \(v(t)=t^2-8t+2\) ft/sec.
Solution
video by Krista King Math |
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Final Answer
\(t=4\) sec
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Find two numbers whose sum is 60 and whose product is maximum.
Problem Statement
Find two numbers whose sum is 60 and whose product is maximum.
Solution
video by The Organic Chemistry Tutor |
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Find two positive numbers whose product is 400 and whose sum is a minimum.
Problem Statement
Find two positive numbers whose product is 400 and whose sum is a minimum.
Solution
video by The Organic Chemistry Tutor |
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Find two numbers whose product is \(-16\) and the sum of whose squares is a minimum.
Problem Statement |
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Find two numbers whose product is \(-16\) and the sum of whose squares is a minimum.
Final Answer |
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\(-4\) and \(4\)
Problem Statement
Find two numbers whose product is \(-16\) and the sum of whose squares is a minimum.
Solution
Break Down The Problem Statement |
First, let's breakdown what the question is telling us. |
Get The Equations |
However, we need an equation (expressions with an equal sign) in order for the problem to make sense. So we just assign this last expression to some variable, let's call it \(z\). So now we have our two equations. |
Determine What We Need To Do To Solve The Problem |
We are asked to minimize this last expression and to minimize we need to take the derivative. However, we have two variables in the equation. So which variable do we need to take the derivative of? Well, to help out we can use the first equation to remove one of the variables from the second equation. When we do that, the second equation will then have only one variable. Let's do that. |
Minimize |
Now rewrite the second term with the exponent term in the numerator \( z = x^2 + 256x^{-2} \). |
Finish Solving |
Now that we know \(x=4\) we can use the first equation, \(xy=-16\), to solve for \(y\) |
So how do we know that this is a minimum and not a maximum? We need to check our answer. There are several ways to do that.
1. We could plot the last equation we had before took the derivative.
2. We could use the first or second derivative test.
3. We could test points on either side of \(x=4\) to make sure they result in larger values in our expressions.
As usual, check with your instructor to see what they require.
Final Answer
\(-4\) and \(4\)
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Intermediate |
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We want an open top rectangular box with volume 10m^{3}. The length of its base is twice the width. Material costs are $10/m^{2} for the base and $6/m^{2} for the sides. Find the minimum cost of the box.
Problem Statement
We want an open top rectangular box with volume 10m^{3}. The length of its base is twice the width. Material costs are $10/m^{2} for the base and $6/m^{2} for the sides. Find the minimum cost of the box.
Solution
video by Khan Academy |
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Really UNDERSTAND Calculus
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