Question

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.

Answer 1

There is a decrease of 0.75 sales per day

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