Suppose that in a certain, company, the relationship between the price per unit p of its product and the weekly sales volume y, in thousands of dollars, is given by . Solve this differential equation if when .

Question

Suppose that in a certain, company, the relationship between the price per unit p of its product and the weekly sales volume y, in thousands of dollars, is given by  . Solve this differential equation if  when  . ​

Quizplus · Accepted Answer

The given differential equation is:
dy/dp = (1/py)(p-10)(50-y)
Separating variables and integrating both sides, we get:
- ln|p-10| + 2ln|y| = ln|C|
Where C is the constant of integration.
Rewriting this in terms of p and y, we get:
y^2/(p-10) = C
Using the initial condition y(10) = 50, we get:
C = 2500
So the equation is:
y^2/(p-10) = 2500
Solving for y, we get:
y = ±50√(p-10)
Since the sales volume y cannot be negative, we take the positive sign, so:
y = 50√(p-10)
Now, when p=20, y=100, so we have:
100 = 50√(20-10)
Solving for the constant of integration, we get:
C = 2500
Substituting this into our equation, we get:
y^2/(p-10) = 2500
Simplifying, we get:
y = 50√(p-10)
So the solution is y = 50√(p-10), which is equivalent to answer choice B.

Suppose That in a Certain, Company, the Relationship Between the Price