Question

find the rate of change of total profit, in dollars, with respect to time where
$r(x)=4x$
and
$c(x)=0.04x^{2}+0.4x + 50$,
when $x = 21$
and $\frac{dx}{dt}=11$.

Explanation:

Step1: Find the profit function

The profit function \( P(x) \) is given by \( P(x)=R(x)-C(x) \).
Substituting \( R(x) = 4x \) and \( C(x)=0.04x^{2}+0.4x + 50 \), we get \( P(x)=4x-(0.04x^{2}+0.4x + 50)=- 0.04x^{2}+3.6x - 50 \).

Step2: Differentiate the profit function with respect to time \( t \)

Using the chain - rule \( \frac{dP}{dt}=\frac{dP}{dx}\cdot\frac{dx}{dt} \).
First, find \( \frac{dP}{dx} \):
\( \frac{dP}{dx}=\frac{d}{dx}(-0.04x^{2}+3.6x - 50)=-0.08x + 3.6 \).

Step3: Substitute \( x = 21 \) and \( \frac{dx}{dt}=11 \)

When \( x = 21 \), \( \frac{dP}{dx}=-0.08\times21+3.6=-1.68 + 3.6 = 1.92 \).
Since \( \frac{dP}{dt}=\frac{dP}{dx}\cdot\frac{dx}{dt} \), and \( \frac{dx}{dt}=11 \), then \( \frac{dP}{dt}=1.92\times11 \).

Step4: Calculate the final result

\( \frac{dP}{dt}=1.92\times11 = 21.12 \).

Answer:

\( 21.12 \)