Question

The demand equation for rubies at Royal Ruby Retailers is q + 3/4p = 80 where q is the number of rubies RRR can sell per week at p dollars per ruby. RRR finds that the demand for its rubies is currently 15 rubies per week and is dropping at a rate of one ruby per week. How fast is the price changing?

Answer 1

To find the rate at which the price is changing when the demand for rubies is decreasing by one ruby per week, we use related rates by differentiating the demand equation implicitly and solving for dp/dt, getting an increase of 4/3 dollars per week.

Step-by-step explanation:

To determine how fast the price p is changing, we need to use related rates, which is a mathematical method involving implicit differentiation. Given the demand equation q + (3/4)p = 80 and the fact that the demand q is decreasing at a rate of -1 ruby per week (which we can denote as dq/dt = -1), we are asked to find the rate at which the price p is changing, represented as dp/dt.

First, we implicitly differentiate the demand equation with respect to time t:

dq/dt + (3/4)dp/dt = 0

Since we have dq/dt = -1, we can substitute it into the equation:

-1 + (3/4)dp/dt = 0

Solving for dp/dt:

(3/4)dp/dt = 1

dp/dt = 4/3

Thus, the price p is increasing at a rate of 4/3 dollars per week.