To determine To find: The value of R x ( 10 , 5 ) and P x ( 10 , 5 ) . Explanation Given: Let x and y are the weekly demand of 10-speed bicycle and 3-speed bicycles. The price of 10-speed bicycle is $ p . The price of 3-speed bicycle is $ q . The weekly demand functions are: p = 230 − 9 x + y q = 130 + x − 4 y The cost function C ( x , y ) = 200 + 80 x + 30 y . Calculation: The revenue function is R ( x , y ) = x p + y q . Substitute p = 230 − 9 x + y and q = 130 + x − 4 y in R ( x , y ) = x p + y q , R ( x , y ) = x ( 230 − 9 x + y ) + y ( 130 + x − 4 y ) = 230 x − 9 x 2 + x y + 130 y + x y − 4 y 2 = − 9 x 2 + 2 x y − 4 y 2 + 230 x + 130 y Thus, the revenue function is R ( x , y ) = − 9 x 2 + 2 x y − 4 y 2 + 230 x + 130 y . Obtain the partial derivative of the revenue function with respect to x . That is, treating y as a constant and differentiate the function with respect to x . R x ( x , y ) = ∂ ∂ x ( − 9 x 2 + 2 x y − 4 y 2 + 230 x + 130 y ) = ∂ ∂ x ( − 9 x 2 ) + ∂ ∂ x ( 2 x y ) − ∂ ∂ x ( 4 y 2 ) + ∂ ∂ x ( 230 x ) + ∂ ∂ x ( 130 y ) = ( − 9 ) ∂ ∂ x ( x 2 ) + ( 2 y ) ∂ ∂ x ( x ) − ∂ ∂ x ( 4 y 2 ) + ( 230 ) ∂ ∂ x ( x ) + ∂ ∂ x ( 130 y ) Simplify the terms and obtain the partial derivatives, R x ( x , y ) = − 9 ( 2 x ) + 2 y ( 1 ) − 0 + 230 ( 1 ) + 0 = − 18 x + 2 y + 230 Therefore, the partial derivative of the function with respect to x is R x ( x , y ) = − 18 x + 2 y + 230 . Substitute 10 for x and 5 for y in R x ( x , y ) = − 18 x + 2 y + 230 , R x ( 10 , 5 ) = − 18 ( 10 ) + 2 ( 5 ) + 230 = − 180 + 10 + 230 = 240 − 180 = 60 Therefore, the partial derivative of the revenue function with respect to x is R x ( 10 , 5 ) = 60

Pearson eText for College Mathematics for Business, Economics, Life Sciences, and Social Sciences -- Instant Access (Pearson+)

14th Edition

ISBN: 9780137553341

Author: Raymond Barnett, Michael Ziegler

Publisher: PEARSON+

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Chapter 14.2, Problem 88E

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To find: The value of Rx(10,5) and Px(10,5).

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Chapter 14.2, Problem 87E

Chapter 14.2, Problem 89E

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Pearson eText for College Mathematics for Business, Economics, Life Sciences, and Social Sciences -- Instant Access (Pearson+)

Ch. 14.1 - Matched Problem 1 Find C(20, 10) for the cost...Ch. 14.1 - Prob. 2MP Ch. 14.1 - Prob. 3MP Ch. 14.1 - Prob. 4MP Ch. 14.1 - Prob. 5MP Ch. 14.1 - Prob. 6MP Ch. 14.1 - Prob. 7MP Ch. 14.1 - Prob. 1ED Ch. 14.1 - Prob. 1E Ch. 14.1 - Prob. 2E

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