Suppose that the price p, in dollars, and the number of sales, x, of a certain item follow the equation 5p+5x+3px=71. Suppose also that p and x are both functions of time, measured in days. Find the rate at which x is changing when x=3, p=4, and dp/dt=1.4. The rate at which x is changing is [ ] sales per day

decrease of 1.153 sales per day

Step-by-step explanation:

Given:-

- The price of item = p

- The number of sales = x

- The relationship between "p" and "x" is given below:

5p+5x+3px=71

Find:-

Find the rate at which x is changing when x=3, p=4, and dp/dt=1.4. The rate at which x is changing is [ ] sales per day

Solution:-

- Take the time derivative (d/dt) of the entire given expression and apply chain rule on d/dt (3px). Since both "p" and "x" are only functions of time "t":

d/dt (5p+5x+3px=71)

5*dp/dt + 5*dx/dt + 3 * (x*dp/dt + p*dx/dt) = 0

- Use the given values x=3, p=4, and dp/dt=1.4 to determine dx/dt:

5*1.4 + 5*dx/dt + 3*3*1.4 + 3*4*dx/dt = 0

17dx/dt = - 19.6

dx/dt = - 1.153 sales per day

- There is a decrease of 1.153 sales per day.