Suppose the daily total cost (in dollars) of manufacturing [tex]$x$[/tex] televisions is
[tex]$C(x)=0.0005 x^3-0.07 x^2+160 x+6000$[/tex]
What is the marginal cost when [tex]$x=500$[/tex]? What is the actual cost incurred in manufacturing the 501st television?
A. [tex]$262.00, $262.41$[/tex]
B. [tex]$465.00, $465.68$[/tex]
C. [tex]$262.00, $262.52$[/tex]
D. [tex]$465.00, $465.93$[/tex]
Answer (2)
Find the marginal cost function by taking the derivative of the total cost function: C ′ ( x ) = 0.0015 x 2 − 0.14 x + 160 .
Evaluate the marginal cost at x = 500 : C ′ ( 500 ) = 465 .
Calculate the total cost of producing 500 and 501 televisions, C ( 500 ) and C ( 501 ) , respectively.
Find the actual cost of producing the 501st television by subtracting C ( 500 ) from C ( 501 ) , which is approximately 465.68. T h e f ina l an s w er i s \boxed{{$465.00, $465.68}}$.
Explanation
Problem Setup We are given the total cost function for manufacturing x televisions:
C ( x ) = 0.0005 x 3 − 0.07 x 2 + 160 x + 6000
We need to find the marginal cost when x = 500 and the actual cost incurred in manufacturing the 501st television.
Finding the Marginal Cost Function The marginal cost is the derivative of the total cost function with respect to x . So, we need to find C ′ ( x ) .
C ′ ( x ) = d x d ( 0.0005 x 3 − 0.07 x 2 + 160 x + 6000 )
Using the power rule, we get:
C ′ ( x ) = 0.0015 x 2 − 0.14 x + 160
Calculating Marginal Cost at x=500 Now, we need to find the marginal cost when x = 500 . We plug in x = 500 into the marginal cost function:
C ′ ( 500 ) = 0.0015 ( 500 ) 2 − 0.14 ( 500 ) + 160
C ′ ( 500 ) = 0.0015 ( 250000 ) − 70 + 160
C ′ ( 500 ) = 375 − 70 + 160
C ′ ( 500 ) = 465
So, the marginal cost when x = 500 is $465.
Calculating Actual Cost of 501st TV The actual cost of manufacturing the 501st television is the difference between the total cost of manufacturing 501 televisions and the total cost of manufacturing 500 televisions:
Actual Cost = C ( 501 ) − C ( 500 )
C ( 500 ) = 0.0005 ( 500 ) 3 − 0.07 ( 500 ) 2 + 160 ( 500 ) + 6000
C ( 500 ) = 0.0005 ( 125000000 ) − 0.07 ( 250000 ) + 80000 + 6000
C ( 500 ) = 62500 − 17500 + 80000 + 6000
C ( 500 ) = 131000 − 17500 = 131000 − 17500 = 113500
C ( 501 ) = 0.0005 ( 501 ) 3 − 0.07 ( 501 ) 2 + 160 ( 501 ) + 6000
C ( 501 ) = 0.0005 ( 125751501 ) − 0.07 ( 251001 ) + 80160 + 6000
C ( 501 ) = 62875.7505 − 17570.07 + 80160 + 6000
C ( 501 ) = 149035.7505 − 17570.07 = 131465.6805
Actual Cost = 131465.6805 − 131000 = 465.6805
So, the actual cost incurred in manufacturing the 501st television is approximately $465.68.
Final Answer The marginal cost when x = 500 is $465, and the actual cost incurred in manufacturing the 501st television is approximately $465.68.
Therefore, the correct answer is:
$465.00 , $465.68
Examples
In business, understanding marginal cost is crucial for making informed decisions about production levels and pricing strategies. For example, a company can use marginal cost analysis to determine the optimal production quantity that maximizes profit. By comparing the marginal cost of producing one more unit with the marginal revenue generated by selling that unit, businesses can decide whether to increase or decrease production. This concept is also vital in cost accounting, helping businesses allocate resources efficiently and make strategic investment decisions.
The marginal cost when x=500 is $465.00, and the actual cost of manufacturing the 501st television is approximately $465.68. Therefore, the correct choice is D: $465.00, $465.68.
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