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β¨Unit 5

2 min readβ’june 8, 2020

Sumi Vora

π₯**Watch: AP Calculus AB/BC - ****Optimization Problems**

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

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

Weβll 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 donβt have to worry about memorizing them.Β

First, letβs try to **find the relationships between all of the variables**, and plugin what we know.Β

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.

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 equationΒ

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

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.π΅Β

- Write a function C(x) that represents the costΒ
- Write a function R(x) that represents revenueΒ
- 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 functionΒ
- Find the number of people that maximizes the profit. What is the maximum profit?Β

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