Answer:
[tex]\dfrac{dx}{dt}=-1.3846$ sales per day[/tex]
Step-by-step explanation:
The price p, in dollars, and the number of sales, x, of a certain item follow the equation: 6p+3x+2px=69
Taking the derivative of the equation with respect to time, we obtain:
[tex]6\dfrac{dp}{dt} +3\dfrac{dx}{dt}+2p\dfrac{dx}{dt}+2x\dfrac{dp}{dt}=0\\$Rearranging$\\6\dfrac{dp}{dt}+2x\dfrac{dp}{dt}+3\dfrac{dx}{dt}+2p\dfrac{dx}{dt}=0\\\\(6+2x)\dfrac{dp}{dt}+(3+2p)\dfrac{dx}{dt}=0[/tex]
When x=3, p=5 and [tex]\dfrac{dp}{dt}=1.5[/tex]
[tex](6+2(3))(1.5)+(3+2(5))\dfrac{dx}{dt}=0\\(6+6)(1.5)+(3+10)\dfrac{dx}{dt}=0\\18+13\dfrac{dx}{dt}=0\\13\dfrac{dx}{dt}=-18\\\dfrac{dx}{dt}=-\dfrac{18}{13}\\\\\dfrac{dx}{dt}=-1.3846$ sales per day[/tex]
The number of sales, x is decreasing at a rate of 1.3846 sales per day.
