Answer Happy

We have the following differential equation.
𝑑𝑦
𝑑𝑥 =(𝑘/𝐿)𝑦(𝐿−𝑦) where k determines how rapidly the function
approaches L
v= 0.92
f=20
L=104
M=0.83
N=12
Just suppose we are in a situation where we have a linear total
cost function (orange line above) 𝐶(𝑥)=
𝑣𝑥+𝑓 with a variable cost v (also the marginal cost 𝑀𝐶̅̅̅̅̅) and a
fixed cost f. Additionally, we have a total
revenue function (blue curve above) R(x) that behaves like a
logistic function, approaching a limiting
value L. What is unusual about this function is that it starts with
a value 𝑅(0)=𝑁 (presumably there is
some revenue independent of sales) and a marginal revenue 𝑀𝑅̅̅̅̅̅
=𝑅′(0)=𝑀.
1. First, use the initial values of y and 𝑑𝑦
𝑑𝑥 and equation (1) to solve for k in terms of L, M, and N.
2. Second, solve for the revenue (y) where profit is a maximum,
using equation (1) to set 𝑀𝑅̅̅̅̅̅ =𝑀𝐶̅̅̅̅̅.
This will involve a quadratic with two solutions. Only one is a
maximum.
3. Third, use antiderivatives on equation (1) to get an implicit
relation between y and x. Make sure
that you also solve for the integration constant, given that
𝑅(0)=𝑁.
4. Use this implicit relationship to solve for the x value that
corresponds to the y value found in step 2
5. Finally calculate the maximum profit by subtracting the cost
(using the x in step 4) from the revenue
you already found in step 2