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Optimization 

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 ProblemsWord 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.

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 \((3,2)\). 
solution 
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. 
solution 
Practice 3 

Calculate the dimensions of a rectangle with area 1000m^{2}, whose perimeter is as small as possible. 
solution 
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 \(\displaystyle{\frac{x^2}{4}+\frac{y^2}{9}=1}\).  
answer 
solution 
Practice 5  

The area of a rectangle is given by \(A(x)=16xx^2\) where x is the length of one of the sides. What is the maximum area of this rectangle?  
answer 
solution 
Practice 6 

Find the dimensions of a rectangle with perimeter 100m where the area is as large as possible. 
solution 
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? 
solution 
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? 
solution 
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? 
solution 
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. 
solution 
Intermediate Problems 
Practice 11 

Calculate the dimensions of an opentopped rectangular box that minimizes the surface area. The volume is 972in^{3} and the length of the bottom is twice as long as the width. 
solution 
Practice 12 

Find the area of the largest rectangle with its base on the xaxis and the other two vertices above the xaxis on the parabola \(y=8x^2\). 
solution 
Practice 13 

Find the area of the largest rectangle that can be inscribed in a semicircle of radius r. 
solution 
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 in^{2}. What dimensions will minimize the size of each page? 
solution 
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. 
solution 
Practice 16  

An open top box with volume 20ft^{3} is twice as long as it is wide. Find the width that minimizes the amount of material required to make the box.  
answer 
solution 
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? 
solution 
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? 
solution 
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. 
solution 
Practice 20  

The total of the length and girth of a box cannot exceed 120in. What dimensions maximize volume if the end is square?  
answer 
solution 
Intermediate Problems 
Practice 21 

Find the largest possible volume of a right circular cylinder that can be inscribed in a sphere with radius r. 
solution 
Practice 22 

Find the maximum volume of a coneshaped 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 
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?  
answer 
solution 
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. 
solution 
Practice 25  

Find two numbers whose product is 81 where the sum of the squares is a minimum.  
answer 
solution 
Practice 26  

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.  
answer 
solution 
Practice 27  

A company sells x thousand candy bars at \(p(x)=62x/12\) cents per bar. How many bars should they sell to maximize revenue, \(R\), where \(R(x)=x[p(x)]\)?  
answer 
solution 
Practice 28  

Find the time when velocity is minimum if velocity is given by \(v(t)=t^28t+2\) ft/sec.  
answer 
solution 
Intermediate Problems 
Practice 29 

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 