Question

Suppose that the price p (in dollars) of a product is given by the demand function p = (18,000 − 60x) / (400 − x) where x represents the quantity demanded and x < 300. f the daily demand is decreasing at a rate of 100 units per day, at what rate (in dollars per day) is the price changing when the price per unit is $30?

Answer 1

To find the rate at which price changes with respect to time when the price is $30, we compute the first derivative of the demand function with respect to quantity, and multiply it by the given rate at which the demand is changing over time.

Step-by-step explanation:

The student's question involves applying derivative concepts from Calculus to determine the rate of change of the price with respect to time.

Given that the demand function is p = (18,000 - 60x) / (400 - x), where x is the quantity demanded, and we know the demand is decreasing at a rate of 100 units per day, we want to find the rate of change of the price when the price per unit is $30.

To find this rate of change, we calculate the first derivative of the price function with respect to x, denote it as dp/dx, and then multiply it by the rate of change of x with respect to time (dx/dt), which is given as -100 units.

This will yield the rate of change of the price with respect to time, or dp/dt, when the price is $30. You will need to solve the initial equation for x when p is $30 beforehand to insert the correct value of x into dp/dx.