Explanation Given information: It is provided that the equation of demand of tuna in a small town is p q 1.5 = 50 , 000 where, q represents the total number of pounds required for tuna which have been sold in one month at the price of p dollars per pound and the town’s fisheries finds that the demand of tuna is now 900 pounds per month and is increasing by the rate of 100 pounds per month each month. Formula used: The steps for solving the related rate problem are: Step1: List the changing and related quantities followed by restating the problem in rate of change of terms. Step 2: Rewrite the problem using for the changing quantities for their derivatives. Step 3: If possible draw the diagram of the changing quantities and find an equation and relate the change in quantities. Step 4: Differentiate with respect to the equation (s) in order to get rate of change in relating quantity. Step 5: Now substitute the values of the provided quantity and their derivative into the derived equation and solve them to get answer. Calculation: Consider the steps of the formula: The changing quantities are the numbers q of pounds of tuna that can be sold in one month, the price p in dollars per pound. Since, the demand for tuna is now 900 pounds per month and is increasing by the rate of 100 pounds per month each month. Thus, the rate of change in demand is 100 pounds per month each month when demand is 900 pounds per month. Thus, d q d t = 100 when q = 900 Since, the rate at which the price is changing is to be determined. Thus, the above sentence can be rephrased as: What is the rate of change of p . Thus, d p d t = ? Consider the demand equation: p q 1.5 = 50 , 000 Substitute, 900 for q in above equation to calculate the p as: p ( 900 ) 1.5 = 50 , 000 p = 50 , 000 27000 p = 50 27 Differentiate the above equation of relationship with respect to t as: d d t ( p q 1.5 ) = d d t ( 50 , 000 ) Consider the product rule of differentiation as: ( a b ) ′ = a ′ b + a b ′ Substitute p for a and q 1

Student Solutions Manual for Waner/Costenoble's Finite Math and Applied Calculus, 7th

7th Edition

ISBN: 9781337275972

Author: Waner, Stefan; Costenoble, Steven

Publisher: Cengage Learning

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Chapter 12.5, Problem 31E

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To calculate: The rate at which the price are changing if the equation of demand of tuna in a small town is pq1.5=50,000 where, q represents the total number of pounds required for tuna which have been sold in one month at the price of p dollars per pound and the town’s fisheries finds that the demand of tuna is now 900 pounds per month and is increasing by the rate of 100 pounds per month each month.

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Chapter 12.5, Problem 30E

Chapter 12.5, Problem 32E

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Solution Summary: The author calculates the rate at which the price is changing if the equation of demand of tuna in a small town is pq1.5=50,000.

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