Explanation Given Information: The company manufactures cylindrical cans with closed tops and volume of 2 cubic centimeters. The metal used to manufacture the cans cost $ 2 per square centimeter for the sides and $ 3 per square centimeter for the top and bottom. Formula used: The Steps for solving the optimization problems: Step 1: Identify the unknowns with the help of diagrams. Step 2: Identify which quantity is to be maximized and which is to be minimized. Step 3: Identify the constraint, the equations relating variables or inequalities expressing limitations on the value of the variables. Step 4: The optimization problem statement. Step 5: Eliminate the extra variables, solve the constraint for one unknown and substitute in the objective function. Step 6: The calculation of the absolute maximum or absolute minimum of the objective function. Calculation: Consider the formula: Step 1: Identify the unknowns. There are two unknowns radius r and height h . Step 2: Identify the objective functions: Take the height of drum as h and the radius of the base as r to get: C = 3 ( 2 π r 2 ) + 2 ( 2 π r h ) C = 3 π r 2 + 4 π r h Step 3: Identify the constraint: V = π r 2 h 2 = π r 2 h Step 4: State the optimization problem: The given objective function is given by: Minimize the objective function. C = 3 π r 2 + 4 π r h Step 5: State and solve the given optimization problem: The objective function is given by: C = 3 π r 2 + 4 π r h The constraint is given by: π r 2 h = 2 Solving the constraint for h to get: h = 2 π r 2 Substituting the value of h as 2 π r 2 in the objective function to get: C = 3 π r 2 + 4 π r h C = 3 π r 2 + 4 π r ( 2 π r 2 ) C = 3 π r 2 + 8 r Thus, 0 < r < ∞ Differentiating both sides of the equation to get: C ′ ( r ) = 6 π r − 8 r 2 Locating stationary points to get: C ′ ( r ) = 0 6 π r − 8 r 2 = 0 Adding 8 r 2 to both sides of the equation to get: 6 π r − 8 r 2 + 8 r 2 = 8 r 2 6 π r = 8 r 2 Cross-Multiplying to get: π r 3 = 8 6 r 3 = 8 6 π Taking cube root of both sides of the equation to get: r = ( 8 6 π ) 1 3 r ≈ 0

Finite Mathematics and Applied Calculus (MindTap Course List)

7th Edition

ISBN: 9781337274203

Author: Stefan Waner, Steven Costenoble

Publisher: Cengage Learning

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Chapter 12.2, Problem 42E

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To calculate: The dimensions of the drum to minimize the cost of metal and the ratio of height and radius. The company manufactures cylindrical cans with closed tops and volume of 2 cubic centimeters. The metal used to manufacture the cans cost $2 per square centimeter for the sides and $3 per square centimeter for the top and bottom.

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Chapter 12.2, Problem 41E

Chapter 12.2, Problem 43E

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Solution Summary: The author explains how the company manufactures cylindrical cans with closed tops and volume of 2 cubic centimeters.

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