Question

A price p (in dollars) and demand x (in items) for a product are related by 2x² −7xp+60p² =25,600. 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 with respect to time. (Round your answer to four decimal places.) The rate of change of demand with respect to time is × items/month ∨.

Answer 1

The rate of change of the demand with respect to time is 22.6 items/month.

Step-by-step explanation:

To find the rate of change of the demand with respect to time, we need to first differentiate the equation of the demand curve with respect to time. Differentiating 2x² - 7xp + 60p² = 25,600:

4x(dx/dt) - 7p(dx/dt) - 7x(dp/dt) + 120p(dp/dt) = 0

Next, we substitute the given values:

4x(dx/dt) - 7(20)(2) - 7x(dp/dt) + 120(20)(2) = 0

At this point, we can solve for dx/dt:

4x(dx/dt) - 280 - 14x(dp/dt) + 4800 = 0

(4x - 14x)(dx/dt) = (280 - 4800)

(-10x)(dx/dt) = -4520

dx/dt = 4520 / (-10x)

Now, we can use the given information that the price is increasing at a rate of 2 dollars per month when the price is 20 dollars. Substitute these values into the equation:

dx/dt = 4520 / (-10(20))

dx/dt = 4520 / (-200) = 22.6 items/month