Suppose that the price $p$, in dollars, and the number of sales, $x$, of a certain item are related by $4 p+3 x+3 p x=46$. If $p$ and $x$ are both functions of time, measured in days. Find the rate at which $x$ is changing when $x=2, p=4$, and $\frac{d p}{d t}=1.2$.
The rate at which $\mathrm{x}$ is changing is (Round to the nearest hundredth as needed.)
Answer
\(\boxed{-0.80}\) is the rate at which $x$ is changing when $x=2$, $p=4$, and $\frac{dp}{dt}=1.2$.
Step 1 ：Differentiate the given equation with respect to time $t$ to get an equation involving $\frac{dp}{dt}$, $\frac{dx}{dt}$, $p$, and $x$.
Step 2 ：The differentiated equation is $4\frac{dp}{dt}+3\frac{dx}{dt}+3p\frac{dx}{dt}+3x\frac{dp}{dt}=0$.
Step 3 ：Substitute the given values $x=2$, $p=4$, and $\frac{dp}{dt}=1.2$ into the differentiated equation.
Step 4 ：This gives $4(1.2)+3\frac{dx}{dt}+3(4)(2)+3(2)(1.2)=0$.
Step 5 ：Solve this equation for $\frac{dx}{dt}$ to get $\frac{dx}{dt}=-0.80$.
Step 6 ：\(\boxed{-0.80}\) is the rate at which $x$ is changing when $x=2$, $p=4$, and $\frac{dp}{dt}=1.2$.
