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Optimizing Simple Systems

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  • 0:40 Visualize It
  • 1:04 Define the Problem
  • 1:21 Write an Equation
  • 2:44 Find the Min/Max
  • 4:29 Answer the Question
  • 4:54 Lesson Summary
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Lesson Transcript
Instructor: Sarah Wright
Optimization problems may seem overwhelming, but they can actually be quite simple. In this lesson, learn how to use a handy five-step formula to tackle these daunting problems.

Five Steps to Solve Optimization Problems

Remember that optimization problems are everywhere, but we have a five-step way to solve optimization problems. We must visualize the problem, define the problem, write an equation for the problem, find the minimum or maximum for the problem and then answer the question. With these steps, we can solve most optimization problems.

The rectangular pen has a height of h, a width of w, and a perimeter of 100 meters
visualize it

Let's try one. What is the maximum amount of land that you can enclose in a rectangular pen that has a perimeter of 100m?

Step 1: Visualize It

The first thing we need to do is visualize it. Let's draw this out; I have a rectangular pen. Let's say it's going to hold my turtle (he needs a lot of room). Here's my rectangular pen: it's got a height of h, a width of w and a perimeter of 100m. I don't know what h and w are since they weren't given to me. Instead, I get to pick those to solve this problem.

Step 2: Define the Problem

Next is the second step, define the problem. I need to maximize the pen area. I'm constrained by my perimeter which has to equal 100m. I don't have any other constraints, so I need to maximize this area which is really the product of my width and height of the pen.

Step 3: Write an Equation

Step 3 involves writing an equation for solving the optimization problem
Optimizing Simple System Step 3

Let's write this as an equation - step three. Area is height times the width, but I don't know what the height or the width is. I do know that the perimeter of this rectangle is 2(height) + 2(width), and the perimeter has to be 100m. When we plug that in, we get 100 = 2h + 2w. Again, area = hw. I'm almost there, but I really want one equation. I don't know how to optimize two equations. Besides, I have three unknowns here. I'm trying to maximize the area, but I've got h and w that can change.

Let's get rid of one of those. Let's solve the perimeter equation for width. So, I've got 100 = 2h + 2w, which I can write as 50 = h + w (I've just divided everything by 2). I can solve that by subtracting h from both sides and I end up with w = 50 - h. Now I can plug my width into my area equation, so that my area just depends on the height. Now, I've got one equation, A = h(50 - h), where h is my height and A is my area.

Step 4: Find the Minimum or Maximum

Find the minimum or maximum is step four of our optimization process. So, we'll write out the area, A = h(50 - h), or A = 50h - hˆ2. I'm going to find the derivative of the area with respect to h. When I differentiate both sides, I get dA/dh = the derivative of the right hand side with respect to h, which is 50 - 2h. Okay, then I can find the critical point of this function by setting that derivative equal to zero (dA/dh = 0). When it's 0, h = 25.

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