Active Calculus - Gvsu - Grand Valley State University

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82 3.4. APPLIED OPTIMIZATIONActivity 3.13.A trough is being constructed by bending a 4 × 24 (measured in feet) rectangular piece of sheetmetal. Two symmetric folds 2 feet apart will be made parallel to the longest side of the rectangleso that the trough has cross-sections in the shape of a trapezoid, as pictured in Figure 3.6. Atwhat angle should the folds be made to produce the trough of maximum volume?θ1 12Figure 3.6: A cross-section of the trough formed by folding to an angle of θ.⊳
3.5. RELATED RATES 833.5 Related RatesPreview Activity 3.5. A spherical balloon is being inflated at a constant rate of 20 cubic inches persecond. How fast is the radius of the balloon changing at the instant the balloon’s diameter is 12inches? Is the radius changing more rapidly when d = 12 or when d = 16? Why?(a) Draw several spheres with different radii, and observe that as volume changes, the radius,diameter, and surface area of the balloon also change.(b) Recall that the volume of a sphere of radius r is V = 4 3 πr3 . Note well that in the setting ofthis problem, both V and r are changing as time t changes, and thus both V and r may beviewed as implicit functions of t, with respective derivatives dV drdtanddt .Differentiate both sides of the equation V = 4 3 πr3 with respect to t (using the chain rule onthe right) to find a formula for dVdtthat depends on both r anddrdt .(c) At this point in the problem, by differentiating we have “related the rates” of change ofV and r. Recall that we are given in the problem that the balloon is being inflated at aconstant rate of 20 cubic inches per second. Is this rate the value of dr dVdtordt ? Why?(d) From part (c), we know the value of dVdtat every value of t. Next, observe that when the diameterof the balloon is 12, we know the value of the radius. In the equation dVdrdt= 4πr2dt ,substitute these values for the relevant quantities and solve for the remaining unknownquantity, which is drdt. How fast is the radius changing at the instant d = 12?(e) How is the situation different when d = 16? When is the radius changing more rapidly,when d = 12 or when d = 16?⊲⊳
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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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Active Calculus - Gvsu - Grand Valley State University

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