Question

Bacteria quadruples on a daily basis. After the first day, there were 50 bacteria. Using the general exponential equation a1=a1⋅r1−1
, what value would represent a1
?(1 point)

Answer 1

In the given situation, we know that the bacteria quadruples daily, which indicates an exponential growth pattern.

The equation you referred to, \( a_1 = a_1 \cdot r^{t-1} \), can be interpreted in the context of exponential growth as follows:

- \( a_1 \) is the initial amount of bacteria.
- \( r \) is the growth factor (in this case, since the bacteria quadruples daily, \( r = 4 \)).
- \( t \) is the time in days.

Since you mentioned that there were 50 bacteria after the first day, that means:

- After 1 day, the bacteria count is 50.
- Therefore, we can denote \( a_1 \) as the initial amount of bacteria at day 0.

To find \( a_1 \), we can reverse-engineer from the observation after the first day:

1. Let \( a_1 \) be the number of bacteria at day 0 (which we want to find).
2. After 1 day (day 1), the number of bacteria is:

\[
a_1 \cdot r^{1-1} = a_1 \cdot r^0 = a_1 \cdot 1 = a_1
\]

3. On day 2, the number of bacteria would be:

\[
a_1 \cdot r^{2-1} = a_1 \cdot r = a_1 \cdot 4
\]

4. Since you're mentioning that there were 50 bacteria after the first day, that means:

\[
a_1 \cdot 4 = 50
\]

From this, we can solve for \( a_1 \):

\[
a_1 = \frac{50}{4} = 12.5
\]

So, the value that would represent \( a_1 \) (the initial number of bacteria at day 0) would be **12.5**.