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.

Question

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 - x^2)

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

thanks

Is anybody going to help me with this problem?

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 - x^2)

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

thanks

Answer 1

To find the marginal profit at a production level of x = 19 widgets, we need to find the derivative of the profit function and evaluate it at x = 19.

The profit function is given by:
P(x) = R(x) - C(x)

Where:
R(x) = Revenue from selling x widgets
C(x) = Total cost of producing x widgets

Given that the fixed costs are 35 billion dollars, we can calculate the variable cost as follows:
Variable Cost per widget = 781 million dollars = 0.781 billion dollars
So, the total cost of producing x widgets is:
C(x) = Fixed Cost + Variable Cost per widget * Number of widgets
C(x) = 35 + 0.781x

Now, let's calculate the revenue function R(x):
R(x) = 0.11(60x - x^2)

To find the profit function P(x), we subtract the total cost function C(x) from the revenue function R(x):
P(x) = R(x) - C(x)
P(x) = 0.11(60x - x^2) - (35 + 0.781x)
P(x) = 6.6x - 0.11x^2 - 35 - 0.781x
P(x) = -0.11x^2 + 5.819x - 35

To find the marginal profit, we need to take the derivative of the profit function with respect to x, and evaluate it at x = 19.

Let's find the derivative of the profit function P(x):
P'(x) = -0.22x + 5.819

Now, we can substitute x = 19 into the derivative to find the marginal profit:
P'(19) = -0.22(19) + 5.819
P'(19) = -4.18 + 5.819

Therefore, the marginal profit at the production level of 19 widgets is:
P'(19) = 1.639 billion dollars per widget.

So, the answer is $1.639 billion per widget.