Can you please help me with this problem? I asked earlier, but nobody was able to help me... I've been working on it for about an hour now.
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
Please help me, I would greatly appreciate this!
Take the derivative of R(x) to get marginal revenue. Solve that for dR(x)/dx for x=19
dR/dx= .11(60-2x)=.11(60-2(19) )
check my thinking.
Sure, I can help you with this problem!
To find the marginal profit at production level x = 19 widgets, we need to calculate the derivative of the revenue function R(x) with respect to x. The derivative will give us the rate of change of revenue with respect to the number of widgets produced, which represents the marginal revenue.
Let's start by finding the derivative of R(x):
R(x) = 0.11(60x - x^2)
To take the derivative, we apply the power rule and the constant multiple rule.
dR(x)/dx = 0.11 * [ d(60x)/dx - d(x^2)/dx ]
For the first term d(60x)/dx, the derivative of 60x with respect to x is 60.
For the second term d(x^2)/dx, the derivative of x^2 with respect to x is 2x.
Substituting these derivatives into the equation, we have:
dR(x)/dx = 0.11 * (60 - 2x)
Now, we can calculate the marginal profit at x = 19 by substituting x = 19 into the derivative:
dR(19)/dx = 0.11 * (60 - 2*19)
Simplifying this expression:
dR(19)/dx = 0.11 * (60 - 38)
dR(19)/dx = 0.11 * 22
dR(19)/dx = 2.42
Therefore, the marginal profit at production level x = 19 widgets is $2.42 billion per widget.
I hope this explanation helps you understand how to solve the problem. If you have any further questions, feel free to ask!