5.11 Solving Optimization Problems

🎥Watch: AP Calculus AB/BC - Optimization Problems

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Example Problem

A cylindrical soda can has the volume V = 32π in^3. What is the minimum surface area of the can? 🥛

First, let’s list all of the variables that we have: volume (V), surface area (S), height (h), and radius (r)

We’ll need to know the volume formula for this problem. Usually, the exam will provide most of these types of formulas (volume of a cylinder, the surface area of a sphere, etc.), so you don’t have to worry about memorizing them. 

First, let’s try to find the relationships between all of the variables, and plugin what we know. 

The question is asking us to minimize the surface area, so we have to take the derivative of S

There is an implicit domain of r>0, because it would be impossible to have a legitimate cylinder with a radius of 0 or a negative number.

Based on the sign chart, we know that 2.52 is the absolute minimum on the implicit domain, so the surface area is minimized when r = 2.52

Since the question is asking what the minimum surface area would be, we simply plug r into the surface area equation 

Practice FRQ

On the AP exam, the examples may seem much more complex, but they will follow the same steps. 

You operate a tour service that offers the following rates for tours: $200 per person if the minimum number of people book the tour (50 people is the minimum), and for each person past 50 people up to a maximum of 80 people, the cost per person is decreased by $2. It costs you $6000 to operate the tour plus $32 per person.💵 

1. Write a function C(x) that represents the cost 

2. Write a function R(x) that represents revenue 

3. Given that profit can be represented by P(x)=R(x)-C(x), write a function that represents profit and state the domain of the function 

4. Find the number of people that maximizes the profit. What is the maximum profit? 

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