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Unit 5

5.11 Solving Optimization Problems

2 min readโ€ขjune 8, 2020

Sumi Vora


๐ŸŽฅWatch: AP Calculus AB/BC - Optimization Problems

Resources:

Example Problem

A cylindrical soda can has the volume V = 32ฯ€ in^3. What is the minimum surface area of the can?ย ๐Ÿฅ›

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

Weโ€™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 donโ€™t have to worry about memorizing them.ย 

First, letโ€™s try to find the relationships between all of the variables, and plugin what we know.ย 

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(522).png?alt=media&token=7e763ac3-b7f7-474b-9b5c-ae6b7308588b

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

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(523).png?alt=media&token=5992cfb9-939e-4e95-90ca-9688b8c24e36

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.

r

0

...

0

...

dS/dr

0

-

2.52

+

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 equationย 

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(525).png?alt=media&token=076cdaac-6605-433d-9ab1-76e94e72c634


Practice FRQ

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

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.๐Ÿ’ตย 

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

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

  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 functionย 

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

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreen%20Shot%202020-05-27%20at%207.22-aNI6PmjEpFJM.png?alt=media&token=f60c8aa5-7543-4470-8a71-ff44854745d9

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