section 2.12: implicit differentiation and relate rates (homework)
score: 80/90 answered: 8/9
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question 9
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a companys revenue from selling x units of an item is given as r = 1600x - 3x². if sales are increasing at a rate of 50 units per day, how rapidly is the revenue increasing per day when 210 units have been sold? (solve this problem using related rates.)
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# Explanation:
## Step1: Differentiate the revenue function
Differentiate $ R = 1600x - 3x^{2}$ with respect to time $ t$ using the sum rule $(u + v)^\prime=u^\prime + v^\prime$ and the constant - multiple rule $(cu)^\prime = cu^\prime$ and the power rule $(x^{n})^\prime=nx^{n - 1}$.
$$
\begin{align*}
\frac{dR}{dt}&=\frac{d}{dt}(1600x)-\frac{d}{dt}(3x^{2})\
&=1600\frac{dx}{dt}-3\times2x\frac{dx}{dt}\
&=(1600 - 6x)\frac{dx}{dt}
\end{align*}
$$
## Step2: Substitute the given values
We are given that $\frac{dx}{dt}=50$ units per day and $x = 210$ units.
Substitute $x = 210$ and $\frac{dx}{dt}=50$ into $\frac{dR}{dt}=(1600 - 6x)\frac{dx}{dt}$.
$$
\begin{align*}
\frac{dR}{dt}&=(1600-6\times210)\times50\
&=(1600 - 1260)\times50\
&=340\times50
\end{align*}
$$

# Answer:
$17000$

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Explanation:

Step1: Differentiate the revenue function

Step2: Substitute the given values

Answer: