Active Calculus - Gvsu - Grand Valley State University

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78 3.4. APPLIED OPTIMIZATION3.4 Applied OptimizationPreview Activity 3.4. According to U.S. postal regulations, the girth plus the length of a parcelsent by mail may not exceed 108 inches, where by “girth” we mean the perimeter of the smallestend. What is the largest possible volume of a rectangular parcel with a square end that can be sentby mail? What are the dimensions of the package of largest volume?Figure 3.4: A rectangular parcel with a square end.(a) Let x represent the length of one side of the square end and y the length of the longer side.Label these quantities appropriately on the image shown in Figure 3.4.(b) What is the quantity to be optimized in this problem? Find a formula for this quantity interms of x and y.(c) The problem statement tells us that the parcel’s girth plus length may not exceed 108inches. In order to maximize volume, we assume that we will actually need the girthplus length to equal 108 inches. What equation does this produce involving x and y?(d) Solve the equation you found in (c) for one of x or y (whichever is easier).(e) Now use your work in (b) and (d) to determine a formula for the volume of the parcel sothat this formula is a function of a single variable.(f) Over what domain should we consider this function? Note that both x and y must bepositive; how does the constraint that girth plus length is 108 inches produce intervals ofpossible values for x and y?(g) Find the absolute maximum of the volume of the parcel on the domain you establishedin (f) and hence also determine the dimensions of the box of greatest volume. Justify thatyou’ve found the maximum using calculus.⊲⊳
3.4. APPLIED OPTIMIZATION 79Activity 3.10.A soup can in the shape of a right circular cylinder is to be made from two materials. Thematerial for the side of the can costs $0.015 per square inch and the material for the lids costs$0.027 per square inch. Suppose that we desire to construct a can that has a volume of 16 cubicinches. What dimensions minimize the cost of the can?(a) Draw a picture of the can and label its dimensions with appropriate variables.(b) Use your variables to determine expressions for the volume, surface area, and cost ofthe can.(c) Determine the total cost function as a function of a single variable. What is the domainon which you should consider this function?(d) Find the absolute minimum cost and the dimensions that produce this value.⊳
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Active CalculusMatt Boelkins, Lead
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Chapter 1Understanding the Derivati
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1.1. HOW DO WE MEASURE VELOCITY? 3A
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1.2. THE NOTION OF LIMIT 51.2 The n
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1.2. THE NOTION OF LIMIT 7Activity
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1.3. THE DERIVATIVE OF A FUNCTION A
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1.3. THE DERIVATIVE OF A FUNCTION A
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1.4. THE DERIVATIVE FUNCTION 131.4
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1.4. THE DERIVATIVE FUNCTION 15rsxx
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1.5. INTERPRETING THE DERIVATIVE AN
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1.5. INTERPRETING THE DERIVATIVE AN
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1.6. THE SECOND DERIVATIVE 21(a) In
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1.6. THE SECOND DERIVATIVE 23• on
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1.6. THE SECOND DERIVATIVE 25ffxxf
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Page 41 and 42: 2.2. THE SINE AND COSINE FUNCTIONS
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Active Calculus - Gvsu - Grand Valley State University

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