Question

Suppose that the price p, in dollars, and number of sales, x, of a certain item follow the equation 5p+4x+2px=60. 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=5, and dpdt=1.5.\

Answer 1

`(dx)/(dt)=-(33)/(28) =-1.1786`

Explanation:

Given that the price p, in dollars, and number of sales, x, of a certain item follow the equation:

5p+4x+2px=60.

Taking partial derivatives

`5(dp)/(dt) +4(dx)/(dt)+2x(dp)/(dt)+2p(dx)/(dt)=0\\(4+2p)(dx)/(dt)=(-5-2x)(dp)/(dt)\\(dx)/(dt)=(-5-2x)/(4+2p) \cdot (dp)/(dt)`

When x=3, p=5, dp/dt=1.5

`(dx)/(dt)=(-5-2(3))/(4+2(5)) \cdot 1.5\\=(-5-6)/(4+10) \cdot 1.5\\=(-11)/(14) \cdot 1.5\\(dx)/(dt)=-(33)/(28) =-1.1786`

Therefore, x is decreasing at a rate of -1.1786.