A price (in dollars) and demand for a product are related by
2x^2-5xp+50p^2=21200
If the price is increasing at a rate of 2 dollars per month when the price is 20 dollars, find the rate of change of the demand.
Rate of change of demand =
To find the rate of change of demand, we need to differentiate the equation with respect to time. Let's denote the price as p (in dollars) and demand as x.
The given equation is:
2x^2 - 5xp + 50p^2 = 21200
Differentiating both sides with respect to time (t) using implicit differentiation, we get:
d/dt (2x^2) - d/dt (5xp) + d/dt (50p^2) = d/dt (21200)
Since we are given that the price is increasing at a rate of 2 dollars per month when the price is 20 dollars, we can substitute these values in the equation.
Using the chain rule, we have:
2 * d/dt (x^2) - 5 * (dx/dt)p - 5x * dp/dt + 100p * dp/dt = 0
Replacing x with the demand variable and dp/dt with the rate of change of price (which is given as 2 dollars per month), we have:
2 * (dx/dt)x - 5 * (dx/dt)p - 5x * 2 + 100p * 2 = 0
Simplifying this equation, we get:
2x * (dx/dt) - 10x - 10xp + 200p = 0
Now, we substitute the current values of p (price) and x (demand) in the equation.
When p = 20 dollars, we have:
2(20) * (dx/dt) - 10(20) - 10(20)x + 200(20) = 0
Simplifying further:
40(dx/dt) - 200 - 200x + 4000 = 0
Rearranging:
40(dx/dt) - 200x = 3200
Now, we can solve for the rate of change of demand (dx/dt):
dx/dt = (200x - 3200) / 40
Now, you can substitute the appropriate value of x (demand) to find the rate of change of demand.