Suppose that the price p, in dollars, and the number of sales, x, of a certain item follow the equation 4 p plus 4 x plus 2 pxequals56. Suppose also that p and x are both functions of time, measured in days. Find the rate at which x is changing when xequals2 , pequals6 , and StartFraction dp Over dt EndFraction equals1.5.

There is a decrease of 0.75 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:

4p+4x+2px=56

Find the rate at which x is changing when x=2, p=6, and dp/dt=1.5 The rate at which x is changing is [ ] sales per day

Take the time derivative (d/dt) of the entire given expression and apply chain rule on d/dt (2px). Since both "p" and "x" are only functions of time "t":

d/dt

4p+4x+2px=56

4*dp/dt + 4*dx/dt + 2 * (x*dp/dt + p*dx/dt) = 0

Use the given values x=2, p=6, and dp/dt=1.5 to determine dx/dt

4*1.5 + 4*dx/dt + 2 * (2*1.5+6*dx/dt) = 0

6+4dx/dt + 2 * (3+6dx/dt) = 0

6+4dx/dt + 6+12dx/dt=0

12+16dx/dt=0

12=-16dx/dt

dx/dt = 12/-16

= - 0.75