A controversial story comes out in the school newspaper. The number of students who have not heard about the story decreases at a rate that is proportional at any time to the number of students who have not heard the story at that time.

There were \[900\] students who had not heard the story initially, and the number of students is divided by \[3\] every \[4\] days.

**How many students have not heard the story after \[7\] days?**
*Round to the nearest student.*

 students

This problem can be modeled using exponential decay. The number of students who have not heard the story decreases over time according to the equation:

\[
N(t) = N_0 \cdot e^{-kt}
\]

where:
- \(N(t)\) is the number of students who haven't heard the story at time \(t\),
- \(N_0\) is the initial number of students who haven't heard the story (900),
- \(k\) is the decay constant,
- \(t\) is the time in days.

### Step 1: Find the decay constant \(k\)

We know that after 4 days, the number of students is divided by 3:

\[
\frac{N_0}{3} = N_0 \cdot e^{-4k}
\]

Dividing both sides by \(N_0\):

\[
\frac{1}{3} = e^{-4k}
\]

Taking the natural logarithm of both sides:

\[
\ln\left(\frac{1}{3}\right) = -4k
\]

\[
k = -\frac{1}{4} \ln\left(\frac{1}{3}\right)
\]

### Step 2: Find the number of students after 7 days

Now, we use the value of \(k\) to find \(N(7)\):

\[
N(7) = 900 \cdot e^{-k \cdot 7}
\]

Substituting the value of \(k\):

\[
N(7) = 900 \cdot e^{-7 \cdot \left(-\frac{1}{4} \ln\left(\frac{1}{3}\right)\right)}
\]

This simplifies to:

\[
N(7) = 900 \cdot e^{\frac{7}{4} \ln\left(\frac{1}{3}\right)}
\]

Using properties of logarithms:

\[
N(7) = 900 \cdot \left(\frac{1}{3}\right)^{\frac{7}{4}}
\]

### Step 3: Compute the result

\[
N(7) = 900 \cdot \left(\frac{1}{3}\right)^{1.75}
\]

Calculate the value:

\[
N(7) \approx 900 \cdot 0.1932 \approx 174
\]

Thus, the number of students who have not heard the story after 7 days is approximately **174** students.

Explore how to model exponential decay in the context of students who have not heard a controversial story in school. Understand the formula N(t) = 900 * (1/3)^(t/4) and compute how many students have not heard the story after 7 days. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation