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AP Calculus AB: 7.3.4 The Can Problem

Mathematics8 CardsCreated 3 months ago

This flashcard set focuses on optimization problems involving cylindrical cans, where the goal is to maximize volume given a fixed surface area or minimize material usage. It covers deriving volume formulas in terms of radius, differentiating to find critical points, and interpreting results to determine the optimal dimensions, supported by real-world application examples.

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The Can Problem

  • Values that make the derivative of a function equal to 0 or undefined are candidates for maxima and minima of the function.

  • The can problem involves maximizing the volume of a cylindrical can constructed from a given quantity of material.

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Key Terms

Term
Definition

The Can Problem

  • Values that make the derivative of a function equal to 0 or undefined are candidates for maxima and minima of the function.

  • ...

note

  • Here you are asked to find the dimensions of a can made from a certain amount of material in a way that maximizes the volume of the can.

The average soda can is made up of 15.625π square inches of material. What dimensions (to the nearest hundredth) would maximize the volume of this cylindrical can?

r = 1.61 inches
h = 3.23inches

What is the maximum volume of an open cylinder (open at both ends)with surface area equal to 16π ft2 and a radius between 1 and 8 feet?

64π ft3

Your average soda can has a height of 5 inches and a circular top and bottom with a diameter of 2.5 inches. What is the surface area of this soda can?

15.625π in2

You are an engineer working for a shipping company who is air mailing huge amounts of gasoline across the country. Your supervisor tells you that the guys upstairs in the suits want to package the gasoline in cylindrical containers that hold 10π cubic feet of gasoline. It is your job to minimize the cost of the packaging materials. What radius should the cylinders have to minimize the amount of packaging material?

^3√5 feet

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AP Calculus AB: 7.3.4 The Can Problem
TermDefinition

The Can Problem

  • Values that make the derivative of a function equal to 0 or undefined are candidates for maxima and minima of the function.

  • The can problem involves maximizing the volume of a cylindrical can constructed from a given quantity of material.

note

  • Here you are asked to find the dimensions of a can made from a certain amount of material in a way that maximizes the volume of the can.

  • You know the surface area of the can as well as the formula for the volume of a cylinder.

  • By solving the surface area for h, you can relate the surface area and the volume. Substituting gives you an equation for volume in terms of the radius of the can.

  • Once you have the formula for the volume, you can
    differentiate.

  • Set the derivative of the volume equal to 0 and solve for r.

  • Since the radius cannot be negative, one of the answers can be rejected.

  • Checking the derivative to the left and the right of the
    maximum candidate will show that the volume increases to the left and decreases to the right. So the candidate does correspond to a maximum.

  • Remember, the question asked for the dimensions of the can, not just the radius. Make sure that you give both dimensions.

The average soda can is made up of 15.625π square inches of material. What dimensions (to the nearest hundredth) would maximize the volume of this cylindrical can?

r = 1.61 inches
h = 3.23inches

What is the maximum volume of an open cylinder (open at both ends)with surface area equal to 16π ft2 and a radius between 1 and 8 feet?

64π ft3

Your average soda can has a height of 5 inches and a circular top and bottom with a diameter of 2.5 inches. What is the surface area of this soda can?

15.625π in2

You are an engineer working for a shipping company who is air mailing huge amounts of gasoline across the country. Your supervisor tells you that the guys upstairs in the suits want to package the gasoline in cylindrical containers that hold 10π cubic feet of gasoline. It is your job to minimize the cost of the packaging materials. What radius should the cylinders have to minimize the amount of packaging material?

^3√5 feet

What is the maximum volume for a closed cylinder with surface area equal to 150 square inches?

141.047 cubic inches

Your average soda can has a height of 5 inches and a circular top and bottom with a diameter of 2.5 inches. What is the volume of this soda can?

7.8125π in3