A company that manufactures sport supplements calculates that its costs and revenue can be modeled by the equations
$$C= 125000 + 0.75 x$$ and $$R = 250 x – \frac{1}{10}{x^2}$$
where $x$ is the number of units of sport supplements produced in $1$ week. Production during one particular week is $1000$ units and is increasing at a rate of $150$ units per week. Find the rates at which the a) cost, b) revenue, and c) profit are changing.
Best Answer
$C=125,000 + 0.75x$ and $R=250x-\frac1{10}x^2$. We are given that $x=1,000$ and $\frac{dx}{dt}=150$ per week.
Using the chain rule, $\frac{dC}{dt}=\frac{dC}{dx}\frac{dx}{dt}$ and $\frac{dR}{dt}=\frac{dR}{dx}\frac{dx}{dt}$.
Thus,
$$\frac{dC}{dt}=0.75 \frac{dx}{dt}=0.75 \times 150$$
and
$$\begin{align} \frac{dR}{dt}&=\left(250-\frac15 x\right)\frac{dx}{dt}\\ &=\left(250-\frac15 (1000)\right) \times 150 \end{align}$$
Since profit is simple revenue minus cost, the rate of change in profit is simply the difference between the rate of change of revenue and the rate of change of cost.