Chapter 12.4, Problem 13E

### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

Chapter
Section

### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

# Suppose that the marginal cost for a certain product is given by MC ¯ = 1.05 ( x + 180 ) 0.05 and marginal revenue is given by MC ¯ = ( 1 / 0.5 x + 4 ) + 208 , where x is in thousands of units and revenue and cost are in thousands of dollars. Suppose further that fixed costs are $150,000 and production is limited to at most 200 thousand units.(a) Find C(x) and R(x).(b) Graph C(x) and R(x) to determine whether a profit can be made.(c) Determine what level of production yields maximum profit and find the maximum profit (or minimum loss). (a) To determine To calculate: The value of C(x) and R(x) where the marginal cost for the product is provided as MC¯=1.05(x+180)0.05 and marginal revenue is provided as MR¯=(10.5x+4)+2.8 where, x is in thousands of units and both revenue and cost are in thousands of dollars. If the fixed cost for the maximum production of 200 thousands units is$200,000.

Explanation

Given information:

The provided marginal cost for the product is given by,

MC¯=1.05(x+180)0.05

And marginal revenue is given by,

MR¯=(10.5x+4)+2.8 where x is in thousands of units.

The provided fixed cost for the maximum production of 200 thousands units is $200,000. Formula used: Total Cost: C(x)=MC¯dx Where MC¯ is defined as the marginal cost. Total Revenue: R(x)=MR¯dx Where MR¯ is defined as the marginal cost. Calculation: Consider the provided marginal cost and marginal revenue, MC¯=1.05(x+180)0.05 and MR¯=(10.5x+4)+2.8 Consider the formula of revenue function, R(x)=MR¯dx Substitute (10.5x+4)+2.8 for MR¯ in above formula to get the value of R(x). R(x)=((10.5x+4)+2.8)dx=((0.5x+4)12+2.8x)dx=2(0.5x+4)120.5+2.8x+C=4(0.5x+4)12+2.8x+C Substitute 0 for x in above equation to get: R(0)=4(0 (b) To determine To graph: The function C(x) and R(x) to find whether the profit can be made where the marginal cost for the product is provided as MC¯=1.05(x+180)0.05 and marginal revenue is provided as MR¯=(10.5x+4)+2.8 where, x is in thousands of units and both revenue and cost are in thousands of dollars. If the fixed cost for the maximum production of 200 thousands units is$200,000.

(c)

To determine

The level of production which provides maximum profit and also calculate the maximum profit or loss. Where the marginal cost for the product is provided as MC¯=1.05(x+180)0.05 and marginal revenue is provided as MR¯=(10.5x+4)+2.8 where, x is in thousands of units.

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