## 17Calculus Derivatives - Optimization

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.

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. ]

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 [9min-17secs]

video by Krista King Math

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.

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

Problem Statement

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

Solution

### 1130 video

video by Krista King Math

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

Problem Statement

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

Solution

### 1123 video

video by PatrickJMT

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.

Problem Statement

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.

Solution

This problem is solved in two videos. The first one just explains the problem while the second one shows his solution.

### 1128 video

video by PatrickJMT

### 1128 video

video by PatrickJMT

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 [10min-21secs]

video by PatrickJMT

Okay, time for some practice problems maximizing area.

Basic Problems

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 $$\displaystyle{\frac{x^2}{4}+\frac{y^2}{9}=1}$$.

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

### 2098 video

video by PatrickJMT

Area = 12 square units

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

### 1121 video

video by Krista King Math

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?

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

### 2100 video

video by PatrickJMT

64 square units

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

### 1122 video

video by PatrickJMT

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

### 1125 video

video by PatrickJMT

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.

### 1135 video

video by Krista King Math

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

### 1136 video

video by Krista King Math

Intermediate Problems

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

### 1131 video

video by Krista King Math

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

### 1124 video

video by PatrickJMT

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

### 1133 video

video by Krista King Math

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

### 1132 video

video by Krista King Math

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

### 1134 video

video by Krista King Math

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.

$$\sqrt{15} \approx 2.466$$ 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

### 2110 video

video by Krista King Math

$$\sqrt{15} \approx 2.466$$ ft

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

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

### 1126 video

video by PatrickJMT

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

Problem Statement

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

Solution

### 1127 video

video by PatrickJMT

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?

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

### 2066 video

video by Krista King Math

40in x 20in x 20in

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

### 1140 video

Intermediate Problems

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

### 1137 video

video by Krista King Math

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

### 1138 video

video by Krista King Math

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?

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

### 2078 video

video by Krista King Math

13cm x 6.5cm

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 [12min-32secs]

video by PatrickJMT

Basic Problems

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

### 1129 video

video by Krista King Math

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.

Problem Statement

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.

$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 750ft2. Fencing costs $5/ft for opposite sides and$8/ft for the other sides. Find the dimensions that minimize the cost.

Solution

### 2067 video

video by Krista King Math

$5/ft side = $$20\sqrt{3}$$ ft;$8/ft side = $$25\sqrt{3}/2$$ ft

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

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

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

### 2108 video

video by Krista King Math

372 thousand candy bars

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

Problem Statement

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

$$t=4$$ sec

Problem Statement

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

Solution

### 2109 video

video by Krista King Math

$$t=4$$ sec

Find two numbers whose product is -16 and the sum of whose squares is a minimum.

Problem Statement

Find two numbers whose product is -16 and the sum of whose squares is a minimum.

Solution

### 1139 video

Intermediate Problems

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.

Problem Statement

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.

Solution

### 1142 video

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