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You CAN Ace Calculus

17calculus > derivatives > optimization

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on this page: ► word problems     ► distances     ► areas     ► volumes     ► other

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.

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 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.

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.

Krista King Math - Optimization - Calculus

Optimizing Distances

Optimizing distances (or lengths) usually involve the use of the Pythagorean Theorem. One idea that you need to keep in mind is that the minimum distance from a point to a line is always perpendicular to the line. This introduces a right triangle which is usually where the Pythagorean Theorem comes in. So you will often have something like \(d=\sqrt{x^2+y^2}\) where \(d\) is a distance.

Here are some practice problems involving length or distance.

Practice 1

Find the point on the line \(2x+y=3\) that is closest to the point \((3,2)\).


Practice 2

Two corridors meet at a right angle. One has width 8 meters, the other has width 27 meters. Find the length of the largest pipe that can be carried horizontally from one hall, around the corner and into the other hall.


Practice 3

Calculate the dimensions of a rectangle with area 1000m2, whose perimeter is as small as possible.


Optimizing Areas

Optimizing area is one of the most common types of problems. Many times they involve rectangles ( \(A=lw\)), triangles (\(A=bh/2\)) 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

Okay, time for some practice problems maximizing area.

Basic Problems

Practice 4

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



Practice 5

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?



Practice 6

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


Practice 7

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?


Practice 8

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?


Practice 9

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?


Practice 10

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.


Intermediate Problems

Practice 11

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.


Practice 12

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 \(y=8-x^2\).


Practice 13

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


Practice 14

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?


Practice 15

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.


Practice 16

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.



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.

Basic Problems

Practice 17

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?


Practice 18

Suppose a cylindrical can is made up of 25.625π square inches of material. What dimensions would maximize the volume of the can?


Practice 19

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.


Practice 20

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



Intermediate Problems

Practice 21

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


Practice 22

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.


Practice 23

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?



Optimizing Other 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.

PatrickJMT - Snell's Law : A Calculus Based Proof

Basic Problems

Practice 24

Find two real numbers with difference 20 and minimum possible product.


Practice 25

Find two numbers whose product is -81 where the sum of the squares is a minimum.



Practice 26

A farmer needs to build a rectangular enclosure of 750ft2. Fencing costs $5/ft for opposite sides and $8/ft for the other sides. Find the dimensions that minimize the cost.



Practice 27

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 \(R(x)=x[p(x)]\)?



Practice 28

Find the time when velocity is minimum if velocity is given by \(v(t)=t^2-8t+2\) ft/sec.



Intermediate Problems

Practice 29

We want an open top rectangular box with volume 10m3. The length of its base is twice the width. Material costs are $10/m2 for the base and $6/m2 for the sides. Find the minimum cost of the box.


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