a manufacturer has determined that the weekly profit from the sale of x items is given by the function…

a manufacturer has determined that the weekly profit from the sale of x items is given by the function below. it is estimated that after t days in any week, x items will have been produced. find the rate of change of profit with respect to time at the end of 6 days. p(x)= - 2x² + 800x - 2000 with x = 0.5t² + 5t answer dp/dt|t = 6 =

a manufacturer has determined that the weekly profit from the sale of x items is given by the function below. it is estimated that after t days in any week, x items will have been produced. find the rate of change of profit with respect to time at the end of 6 days. p(x)= - 2x² + 800x - 2000 with x = 0.5t² + 5t answer dp/dt|t = 6 =

Answer

Explanation:

Step1: Apply chain - rule

By the chain - rule, $\frac{dP}{dt}=\frac{dP}{dx}\cdot\frac{dx}{dt}$. First, find $\frac{dP}{dx}$ and $\frac{dx}{dt}$. Differentiate $P(x)=-2x^{2}+800x - 2000$ with respect to $x$: $\frac{dP}{dx}=\frac{d}{dx}(-2x^{2}+800x - 2000)=-4x + 800$. Differentiate $x = 0.5t^{2}+5t$ with respect to $t$: $\frac{dx}{dt}=\frac{d}{dt}(0.5t^{2}+5t)=t + 5$. So, $\frac{dP}{dt}=(-4x + 800)(t + 5)$.

Step2: Find $x$ when $t = 6$

Substitute $t = 6$ into $x = 0.5t^{2}+5t$. $x=0.5\times6^{2}+5\times6=0.5\times36 + 30=18 + 30=48$.

Step3: Calculate $\frac{dP}{dt}$ at $t = 6$

Substitute $x = 48$ and $t = 6$ into $\frac{dP}{dt}=(-4x + 800)(t + 5)$. $\frac{dP}{dt}=(-4\times48+800)(6 + 5)=(-192 + 800)\times11=608\times11 = 6688$.

Answer:

$6688$