Subjects and resources that you bookmark will appear here.

Sumi Vora

🎥**Watch: AP Calculus AB/BC - ****Optimization Problems**

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.

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 equation

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.💵

Write a function C(x) that represents the cost

Write a function R(x) that represents revenue

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

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

Thousands of students are studying with us for the AP Calculus AB/BC exam.

join nowBrowse Study Guides By Unit

✍️

Free Response Questions (FRQ)

🧐

Multiple Choice Questions (MCQ)

♾

Unit 10: Infinite Sequences and Series (BC Only)

👑

Unit 1: Limits & Continuity

🤓

Unit 2: Differentiation: Definition & Fundamental Properties

🤙🏽

Unit 3: Differentiation: Composite, Implicit & Inverse Functions

👀

Unit 4: Contextual Applications of the Differentiation

✨

Unit 5: Analytical Applications of Differentiation

🔥

Unit 6: Integration and Accumulation of Change

💎

Unit 7: Differential Equations

🐶

Unit 8: Applications of Integration

🦖

Unit 9: Parametric Equations, Polar Coordinates & Vector Valued Functions (BC Only)

Practice your typing skills while reading Solving Optimization Problems

Start Game