Question

A population of cells triples every 2 hours. If there were 150 cells at 10:00 a.m., which equation represents the number of cells, n, at any given time, t, in hours?

Answer 1

To model the growth of the cell population, we can use the formula for exponential growth. In this case, the cells triple every 2 hours.

1. **Identify the initial condition**: At \( t = 0 \) hours (which is 10:00 a.m.), there are \( n(0) = 150 \) cells.

2. **Understand the growth factor**: Since the population triples every 2 hours, the growth factor per hour can be calculated. If the population triples every 2 hours, the growth factor is:
\[
\text{Growth factor} = 3^{\frac{t}{2}}
\]
This represents that after every 2 hours, the population is multiplied by 3.

3. **Combine these elements into an exponential growth equation**: The general formula for exponential growth can be expressed as:
\[
n(t) = n_0 \cdot r^{kt}
\]
where:
- \( n_0 \) is the initial population (150),
- \( r \) is the growth factor per time unit (3 for every 2 hours),
- \( k \) is the number of time periods in hours for the growth factor (with \( k = \frac{1}{2} \) for each hour because it takes 2 hours to triple).

Thus the equation becomes:
\[
n(t) = 150 \cdot 3^{\frac{t}{2}}
\]

This equation gives the number of cells \( n \) after \( t \) hours from 10:00 a.m.