A company that manufactures wigdgets has hired a calculus student to determine the optimal number of widgets to produce each month in order to maximize profit. After a brief review, the student realizes that revenue is determined by
R(n)= __3n^2___ +5
e^0.5n
Where n is the number of widgets produced per month, in hundreds, and revenue is measured in millions of dollars.
Determine the number of widgets that should be produced per month to maximize revenue.
R(n) =
_3n^2___ +5
e^0.5n
Sorry if its written poorly, its a fraction with a +5 after it
In the future you would write the fraction like this
R(n)= (3n^2)/(e^0.5n) + 5
We know that for a maximum of R(n), R ' (n) = 0
using the quotient rule:
R'(n) = (e^0.5n)(6n) - 3n^2(0.5) e^.5n)/(e^.5n)^2 + 0
= 0 for our max
e^.5n (6n - 1.5n^2) = 0
well, e^.5n can never be zero, so
6n - 1.5n^2 = 0
n(6 - 1.5n) = 0
n = 0 ----> that would give us our minimu, we don't care about that
or
n = 6/1.5 = 4
finish it up and state your conclusion
To determine the optimal number of widgets to produce per month in order to maximize revenue, we need to find the value of 'n' that maximizes the revenue function R(n).
Step 1: Take the derivative of the revenue function R(n) with respect to 'n'.
The derivative of R(n) can help us find the critical points, where the slope is equal to zero (revenue is maximized).
To find the derivative, we can use the quotient rule:
dR(n)/dn = [(3n^2 + 5)(0.5e^(0.5n)) - e^(0.5n)(6n)] / (e^(0.5n))^2
Simplifying this equation further, we get:
dR(n)/dn = [(1.5n^2e^(0.5n) + 2.5e^(0.5n)) - 6ne^(0.5n)] / e^n
Step 2: Set the derivative equal to zero and solve for 'n'.
Setting dR(n)/dn = 0, we get:
(1.5n^2e^(0.5n) + 2.5e^(0.5n) - 6ne^(0.5n)) / e^n = 0
Multiply both sides of the equation by e^n to eliminate the denominator:
1.5n^2e^(n) + 2.5e^(n) - 6ne^(n) = 0
Step 3: Solve for 'n'.
Now, we need to solve this equation to find the value of 'n' that maximizes revenue. However, this equation does not have a straightforward algebraic solution.
We can either solve this equation numerically using computational methods or with the help of a graphing calculator or software like Excel, Matlab, or Python.
One approach is to plot the graph of the function y = 1.5n^2 + 2.5 - 6n and find the x-coordinate of the vertex, which will give us the optimal value of 'n' that maximizes revenue.
Alternatively, we can use optimization algorithms or numerical methods like Newton's method or bisection method to find an approximate solution.
Step 4: Analyze the solution.
Once we have found the value of 'n' that maximizes revenue, we can substitute it back into the revenue function R(n) to find the actual maximum revenue achieved.
Note: Since the specific values for 'n' and the revenue function are not provided in the question, we cannot directly determine the optimal number of widgets to produce per month to maximize revenue.