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The rate of change
(dP)/(dt) of the number of people at a carnival is modeled by the following differential equation:

(dP)/(dt)=(2881)/(7178)P(1-(P)/( 536))
At
t=0, the number of people at the carnival is 148 and is increasing at a rate of 43 people per hour. At what value of
P does the graph of
P(t) have an inflection point?
Answer:

The rate of change $\frac{d P}{d t}$ of the number of people at a carnival is modeled by the following differential equation: $\newline$ $\frac{d P}{d t}=\frac{2881}{7178} P\left(1-\frac{P}{536}\right)$ $\newline$ At $t=0$ , the number of people at the carnival is $148$ and is increasing at a rate of $43$ people per hour. At what value of $P$ does the graph of $P(t)$ have an inflection point? $\newline$ Answer:

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Find derivatives using logarithmic differentiation

Full solution

Q. The rate of change

\frac{d P}{d t}

of the number of people at a carnival is modeled by the following differential equation:

\newline

\frac{d P}{d t}=\frac{2881}{7178} P\left(1-\frac{P}{536}\right)

\newline

At

t=0

, the number of people at the carnival is

148

and is increasing at a rate of

43

people per hour. At what value of

P

does the graph of

P(t)

have an inflection point?

\newline

Answer:

Find Second Derivative: To find the inflection point, we need to find the second derivative of $P$ with respect to $t$ and then determine when it changes sign. The given differential equation is: $\newline$ $\frac{dP}{dt} = \frac{2881}{7178}P\left(1 - \frac{P}{536}\right)$ $\newline$ First, we need to find the first derivative of $\frac{dP}{dt}$ with respect to $P$ , which will give us $\frac{d^2P}{dt^2}$ .
Calculate First Derivative: The first derivative of $(\frac{dP}{dt})$ with respect to $P$ is given by the derivative of the right-hand side of the equation with respect to $P$ . We apply the product rule and the chain rule to find this derivative: $\newline$ $(\frac{d^2P}{dt^2}) = \frac{d}{dP} \left[\frac{2881}{7178}P\left(1 - \frac{P}{536}\right)\right]$ $\newline$ Let's denote $\frac{2881}{7178}$ as a constant $k$ for simplicity: $\newline$ $k = \frac{2881}{7178}$ $\newline$ Now, we differentiate: $\newline$ $(\frac{d^2P}{dt^2}) = k * \frac{d}{dP} \left[P - \frac{P^2}{536}\right]$ $\newline$ $(\frac{d^2P}{dt^2}) = k * \left(1 - \frac{2P}{536}\right)$
Set Second Derivative to Zero: To find the inflection point, we set the second derivative equal to zero and solve for $P$ : $0 = k \times \left(1 - \frac{2P}{536}\right)$ Since $k$ is a positive constant, we can divide both sides by $k$ without changing the inequality: $0 = 1 - \frac{2P}{536}$ Now, we solve for $P$ : $\frac{2P}{536} = 1$ $2P = 536$ $P = \frac{536}{2}$ $P = 268$
Solve for Inflection Point: We have found that the second derivative changes sign at $P = 268$ . This is the value of $P$ at which the graph of $P(t)$ has an inflection point.

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