Explanation Given Information: The annual revenue for cellphone uses in China for the duration 2000 − 2005 is given by the equation R ( t ) = 14 t + 24 billion dollars , where t is the time since 2000 . The number of subscribers was about 68 million in 2000 and the number of subscribers increases exponentially with an annual growth constant of 10 % . Formula used: Quotient rule of derivative of differentiable functions, f ( x ) and g ( x ) is, d d x [ f ( x ) g ( x ) ] = f ' ( x ) g ( x ) − f ( x ) g ' ( x ) [ g ( x ) ] 2 Where, g ( x ) ≠ 0 . Constant multiple rule of derivative of function f ( x ) is f ' ( c x ) = c f ' ( x ) where c is constant. The derivative of e raised to a function is d d x e u = e u d u d x . Calculation: Consider that the annual revenue for cellphone use in China for the duration 2000 − 2005 is given by the equation R ( t ) = 14 t + 24 billion dollars , where t is the time since 2000 . The number of subscribers was about 68 million in 2000 and the number of subscribers increases exponentially with an annual growth constant of 10 % . Assume, S ( t ) be the function that represent the annual revenue per subscriber. The number of subscribers is given by, A ( t ) = A e r t Where, A = Initial value t = Time taken r = Growth Constant Consider the formula A ( t ) = A e r t The number of subscribers was 68 million or 0.068 billion in 2000 and the number of subscribers increases exponentially with an annual growth constant of 10 % . So, A = 0.068 r = 10 100 = 0.1 Substitute the values in the equation as follows: A ( t ) = 0.068 e 0.1 t The annual revenue per subscriber, S ( t ) is calculated as dividing R ( t ) by A ( t ) , S ( t ) = 14 t + 24 0.068 e 0.1 t billion dollars per subscriber In the year 2002 , t = 2 Substitute t = 2 in S ( t ) to calculate the annual revenue per subscriber in the year 2002 , S ( 2 ) = 14 ( 2 ) + 24 0.068 e 0.1 ( 2 ) = 52 0.068 e 0.2 = 52 0.083 ≈ 626 Thus, the annual revenue per subscriber in the year 2002 is $ 626 billion per subscriber. The derivative of S ( t ) is equal to the rate of change of the annual revenue. S ′ ( t ) = d d t ( 14 t + 24 0.068 e 0.1 t ) Apply the quotient rule of derivative, S ′ ( t ) = d d t ( 14 t + 24 ) ⋅ ( 0.068 e 0.1 t ) − ( 14 t + 24 ) ⋅ d d t ( 0.068 e 0.1 t ) ( 0.068 e 0.1 t ) 2 = ( d d t ( 14 t ) + d d t ( 24 ) ) ⋅ ( 0

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 11.5, Problem 112E

To determine

To calculate: The formula for the annual revenue per subscriber in year t, the value of the annual revenue per subscriber and its rate of change in 2002 if the annual revenue for cellphone use in China for the duration 2000−2005 is given by the equation R(t)=14t+24 billion dollars, where t is the time since 2000. The number of subscribers was about 68 million in 2000 and the number of subscribers increases exponentially with an annual growth constant of 10%.

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Chapter 11.5, Problem 111E

Chapter 11.5, Problem 113E

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Solution Summary: The author calculates the formula for the annual revenue per subscriber in year t and the rate of change in 2002.

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