SECTION 13.4 ✓ CHECKPOINT ANSWERS Applications of Definite Integrals in Business and Economics 823 (a) 5000e Ϫ0.01t dt 5 (b) (5000e Ϫ0.01t)(e Ϫ0.07t) dt ϭ 5000e Ϫ0.08t dt 0 5 (c) e(0.07)(5) (5000e Ϫ0.01t)(e Ϫ0.07t) dt ϭ e 0.353 5000e Ϫ0.08t dt 0 (a) False Consumer’s surplus uses the demand function, so x1 CS ϭ g(x) dx Ϫ p1x1 (b) False Producer’s surplus uses the supply function, but the formula is x1 PS ϭ p1x1 Ϫ f (x) dx (a) CS ϭ 100 dx Ϫ 90 xϩ1 (b) PS ϭ 90 Ϫ (x ϩ 1) dx | EXERCISES | 13.4 CONTINUOUS INCOME STREAMS Find the total income over the next 10 years from a continuous income stream that has an annual rate of flow at time t given by f (t) ϭ 12,000e 0.01t (dollars per year) Find the total income over the next years from a continuous income stream with an annual rate of flow at time t given by f (t) ϭ 8500e Ϫ0.2t (dollars per year) Suppose that a steel company views the production of its continuous caster as a continuous income stream with a monthly rate of flow at time t given by f (t) ϭ 24,000e 0.03t (dollars per month) Find the total income from this caster in the first year Suppose that the Quick-Fix Car Service franchise finds that the income generated by its stores can be modeled by assuming that the income is a continuous stream with a monthly rate of flow at time t given by f (t) ϭ 10,000e 0.02t (dollars per month) Find the total income from a Quick-Fix store for the first years of operation A small brewery considers the output of its bottling machine as a continuous income stream with an annual rate of flow at time t given by f (t) ϭ 80e Ϫ0.1t in thousands of dollars per year Find the income from this stream for the next 10 years A company that services a number of vending machines considers its income as a continuous stream with an annual rate of flow at time t given by f (t) ϭ 120e Ϫ0.4t in thousands of dollars per year Find the income from this stream over the next years A franchise models the profit from its store as a continuous income stream with a monthly rate of flow at time t given by f (t) ϭ 3000e 0.004t (dollars per month) When a new store opens, its manager is judged against the model, with special emphasis on the second half of the first year Find the total profit for the second 6-month period (t ϭ to t ϭ 12) The Medi Spa franchise has a continuous income stream with a monthly rate of flow modeled by f(t) ϭ 20,000e 0.03t (dollars per month) Find the total income for years through A continuous income stream has an annual rate of flow at time t given by f (t) ϭ 12,000e 0.04t (dollars per year) If money is worth 8% compounded continuously, find the present value of this stream for the next 8 years 10 A continuous income stream has an annual rate of flow at time t given by f (t) ϭ 9000e 0.12t (dollars per year) Find the present value of this income stream for the next 10 years, if money is worth 6% compounded continuously 11 The income from an established chain of laundromats is a continuous stream with its annual rate of flow at time t given by f (t) ϭ 630,000 (dollars per year) If money is worth 7% compounded continuously, find the present value and future value of this chain over the next years 12 The profit from an insurance agency can be considered as a continuous income stream with an annual rate of flow at time t given by f (t) ϭ 840,000 (dollars per year) Find the present value and future value of this agency Copyright 2016 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 824 CHAPTER 13 Definite Integrals: Techniques of Integration over the next 12 years, if money is worth 8% compounded continuously 13 Suppose that a printing firm considers its production as a continuous income stream If the annual rate of flow at time t is given by f (t) ϭ 97.5e Ϫ0.2(tϩ3) in thousands of dollars per year, and if money is worth 6% compounded continuously, find the present value and future value of the presses over the next 10 years 14 Suppose that a vending machine company is considering selling some of its machines Suppose further that the income from these particular machines is a continuous stream with an annual rate of flow at time t given by f (t) ϭ 12e Ϫ0.4(tϩ3) in thousands of dollars per year Find the present value and future value of the machines over the next years if money is worth 10% compounded continuously 15 A 58-year-old couple are considering opening a business of their own They will either purchase an established Gift and Card Shoppe or open a new Wine Boutique The Gift Shoppe has a continuous income stream with an annual rate of flow at time t given by G(t) ϭ 30,000 (dollars per year) and the Wine Boutique has a continuous income stream with a projected annual rate of flow at time t given by W(t) ϭ 21,600e 0.08t (dollars per year) The initial investment is the same for both businesses, and money is worth 10% compounded continuously Find the present value of each business over the next 7 years (until the couple reach age 65) to see which is the better buy 16 If the couple in Problem 15 plan to keep the business until age 70 (for the next 12 years), find each present value to see which business is the better buy in this case CONSUMER’S SURPLUS In Problems 17–26, p and C are in dollars and x is the number of units 17 The demand function for a product is p ϭ 34 Ϫ x If the equilibrium price is $9 per unit, what is the consumer’s surplus? 18 The demand function for a product is p ϭ 100 Ϫ 4x If the equilibrium price is $40 per unit, what is the consumer’s surplus? 19 The demand function for a product is p ϭ 200 (x ϩ 2) If the equilibrium quantity is units, what is the consumer’s surplus? 20 The demand function for a certain product is p ϭ 100 (1 ϩ 2x) If the equilibrium quantity is 12 units, what is the consumer’s surplus? 21 The demand function for a certain product is p ϭ 81 Ϫ x2 and the supply function is p ϭ x ϩ 4x ϩ 11 Find the equilibrium point and the consumer’s surplus there 22 The demand function for a product is p ϭ 49 Ϫ x and the supply function is p ϭ 4x ϩ Find the equilibrium point and the consumer’s surplus there 23 If the demand function for a product is p ϭ 12 (x ϩ 1) and the supply function is p ϭ ϩ 0.2x, find the consumer’s surplus under pure competition 24 If the demand function for a good is p ϭ 110 Ϫ x and the supply function for it is p ϭ Ϫ 56x ϩ 15x 2, find the consumer’s surplus under pure competition 25 A monopoly has a total cost function C ϭ 1000 ϩ 120x ϩ 6x for its product, which has demand function p ϭ 360 Ϫ 3x Ϫ 2x Find the consumer’s surplus at the point where the monopoly has maximum profit 26 A monopoly has a total cost function C ϭ 500 ϩ 2x ϩ 10x for its product, which has demand function p ϭ Ϫ13x Ϫ 2x ϩ 30 Find the consumer’s surplus at the point where the monopoly has maximum profit PRODUCER’S SURPLUS In Problems 27–36, p is in dollars and x is the number of units 27 Suppose that the supply function for a good is p ϭ 4x ϩ 2x ϩ If the equilibrium price is $422 per unit, what is the producer’s surplus there? 28 Suppose that the supply function for a good is p ϭ 0.1x ϩ 3x ϩ 20 If the equilibrium price is $36 per unit, what is the producer’s surplus there? 29 If the supply function for a commodity is p ϭ 10e x 3, what is the producer’s surplus when 15 units are sold? 30 If the supply function for a commodity is p ϭ 40 ϩ 100 (x ϩ 1)2, what is the producer’s surplus at x ϭ 20? 31 Find the producer’s surplus at market equilibrium for a product if its demand function is p ϭ 81 Ϫ x and its supply function is p ϭ x ϩ 4x ϩ 11 32 Find the producer’s surplus at market equilibrium for a product if its demand function is p ϭ 49 Ϫ x and its supply function is p ϭ 4x ϩ 33 Find the producer’s surplus at market equilibrium for a product with demand function p ϭ 12 (x ϩ 1) and supply function p ϭ ϩ 0.2x 34 Find the producer’s surplus at market equilibrium for a product with demand function p ϭ 110 Ϫ x and supply function p ϭ Ϫ 56x ϩ 51x 35 The demand function for a certain product is p ϭ 144 Ϫ 2x and the supply function is p ϭ x ϩ 33x ϩ 48 Find the producer’s surplus at the equilibrium point 36 The demand function for a product is p ϭ 280 Ϫ 4x Ϫ x and the supply function for it is p ϭ 160 ϩ 4x ϩ x Find the producer’s surplus at the equilibrium point Copyright 2016 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it ... value of the machines over the next years if money is worth 10% compounded continuously 15 A 58-year-old couple are considering opening a business of their own They will either purchase an established