A widget factory has fixed costs of 35 billion dollars and variable costs of 781 million dollars per widget. The revenue (in $ billions) from selling x number of widgets is given by the following for x between 0 and 60.
R(x) = 0.11 (60x - x2)
What is the marginal profit (in $ billions per widget) at production level x = 19 widgets? (Give your answer correct to 3 decimal places.)
$ ____ billion per widget
Write down the cost function C(x) and the revenue function R(x). They have already told you that the R(x) function is
R(x) = 0.11(60 x - x^2)
C(x) = 35 + 0.781 x (in $billions)
The marginal profit is R'(x) - C'(x)
The ' notation denotes a derivative.
R'(x) = 6.6 - 0.22 x
Complete the final steps and substitute x = 19 for the marginal cost of selling one more when x = 19.
To find the marginal profit at production level x = 19 widgets, we need to calculate R'(x) - C'(x) and substitute x = 19 into the resulting expression.
First, let's find the derivative of the revenue function R(x):
R(x) = 0.11(60x - x^2)
Differentiating with respect to x, we get:
R'(x) = 0.11(60 - 2x)
Now, let's find the derivative of the cost function C(x):
C(x) = 35 + 0.781x
Differentiating with respect to x, we get:
C'(x) = 0.781
Now, let's substitute x = 19 into R'(x) - C'(x):
R'(19) - C'(19) = 0.11(60 - 2*19) - 0.781
= 0.11(60 - 38) - 0.781
= 0.11(22) - 0.781
= 2.42 - 0.781
= 1.639
Therefore, the marginal profit at production level x = 19 widgets is $1.639 billion per widget.