Question

a rumor is spreading through a high school and follows the logistic function:
$r(t)=\frac{4000}{1 + 799e^{-\frac{1}{4}t}}$
where $r(t)$ is a function of the number of students that know the rumor $t$ days since the
rumor began.
determine $r(0)$: select
what is the meaning of $r(0)$ in the context of this situation? select
what is the carrying capacity of this logistic function? select

Explanation:

Step1: Calculate \( R(0) \)

Substitute \( t = 0 \) into \( R(t)=\frac{4000}{1 + 799e^{-\frac{1}{4}t}} \).
When \( t = 0 \), \( e^{-\frac{1}{4}\times0}=e^{0}=1 \).
So \( R(0)=\frac{4000}{1 + 799\times1}=\frac{4000}{800}=5 \).

Step2: Explain the meaning of \( R(0) \)

In the context of the rumor - spreading situation, \( R(0) \) represents the number of students who knew the rumor at the start (\( t = 0 \), when the rumor began).

Step3: Find the carrying capacity

For a logistic function of the form \( y=\frac{L}{1+Ae^{-kt}} \), the carrying capacity is \( L \).
In the given function \( R(t)=\frac{4000}{1 + 799e^{-\frac{1}{4}t}} \), the carrying capacity is \( 4000 \).

Answer:

• \( R(0)=5 \)
• The meaning of \( R(0) \): The number of students who knew the rumor at the beginning (\( t = 0 \)).
• Carrying capacity: \( 4000 \)