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Part 1
As a town gets​ smaller, the population of its high school decreases by 8​% each year. The senior class has 160 students now. In how many years will it have about 100 ​students? Write an equation. Then solve the equation.
Question content area bottom
Part 1
In tequals

enter your response here years the senior class will have about 100 students.
​(Type an integer or decimal rounded to the nearest hundredth as​ needed.)

Let's start by setting up the equation.

Given:
- The population decreases by \(8\%\) per year.
- The current population of the senior class is \(160\).
- We want to find the number of years, \(t\), when the senior class population will be about \(100\).

### Step 1: Set up the equation

The decrease in population by \(8\%\) each year means we retain \(92\%\) of the population every year. The population after \(t\) years can be represented by:
\[
160 \times (0.92)^t = 100
\]

### Step 2: Solve for \(t\)

1. Divide both sides by \(160\):
   \[
   (0.92)^t = \frac{100}{160}
   \]
   \[
   (0.92)^t = 0.625
   \]

2. Take the natural logarithm (ln) of both sides to solve for \(t\):
   \[
   \ln((0.92)^t) = \ln(0.625)
   \]
   
3. Use the power rule of logarithms, \(\ln(a^b) = b \cdot \ln(a)\):
   \[
   t \cdot \ln(0.92) = \ln(0.625)
   \]

4. Divide both sides by \(\ln(0.92)\):
   \[
   t = \frac{\ln(0.625)}{\ln(0.92)}
   \]

5. Calculate the values:
   \[
   t \approx \frac{-0.470}{-0.0834} \approx 5.64
   \]

### Answer

Rounded to the nearest hundredth, the number of years \(t\) is approximately:
\[
t \approx 5.64 \text{ years}
\]

So, in approximately **5.64 years**, the senior class will have about 100 students.

Would you like more details or have questions on this solution?

Here are five related questions:
1. What would happen if the population decreased by a different percentage, like 10% per year?
2. How would the equation change if the initial senior class population was different?
3. What if we wanted to know when the senior class would reach 50 students?
4. How does the solution change if we use continuous decay instead of annual decay?
5. Can we determine the population for any given year with this model?

**Tip:** When solving exponential decay problems, converting to a logarithmic form is a key step to isolate the exponent.

As a town gets smaller, the population of its high school decreases by 8% each year. The senior class has 160 students now. In how many years will it have about 100 students? Write an equation. Then solve the equation.

Explore a step-by-step solution to determine how many years it will take for a high school class of 160 students to decrease to 100 students, with an 8% yearly decay. Key concepts include exponential decay, algebra, and logarithmic calculations, ideal for Grades 9-12.

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