Question

A specific type of algae in a lake grows at an exponential rate under ideal conditions. The population of the algae doubles every 3 days. The recursive formula for this growth is P(n)=P(n−3)⋅2
𝑃
(
𝑛
)
=
𝑃
(
𝑛
−
3
)
⋅
2
, where P(n)
𝑃
(
𝑛
)
represents the population on the nth day, and P(0)=1,000
𝑃
(
0
)
=
1
,
000
. Determine the number of days it will take for the population to reach at least 500,000. (1 point)
days

Answer 1

To determine how many days it takes for the population of algae to reach at least 500,000, we start by using the given recursive formula and the initial population:

1. We start with \( P(0) = 1000 \).
2. The algae doubles every 3 days, according to the formula \( P(n) = P(n-3) \cdot 2 \).

Using the recursive formula, let's calculate the population for several days:

- **For 0 days:**
\[
P(0) = 1000
\]

- **For 3 days:**
\[
P(3) = P(0) \cdot 2 = 1000 \cdot 2 = 2000
\]

- **For 6 days:**
\[
P(6) = P(3) \cdot 2 = 2000 \cdot 2 = 4000
\]

- **For 9 days:**
\[
P(9) = P(6) \cdot 2 = 4000 \cdot 2 = 8000
\]

- **For 12 days:**
\[
P(12) = P(9) \cdot 2 = 8000 \cdot 2 = 16000
\]

- **For 15 days:**
\[
P(15) = P(12) \cdot 2 = 16000 \cdot 2 = 32000
\]

- **For 18 days:**
\[
P(18) = P(15) \cdot 2 = 32000 \cdot 2 = 64000
\]

- **For 21 days:**
\[
P(21) = P(18) \cdot 2 = 64000 \cdot 2 = 128000
\]

- **For 24 days:**
\[
P(24) = P(21) \cdot 2 = 128000 \cdot 2 = 256000
\]

- **For 27 days:**
\[
P(27) = P(24) \cdot 2 = 256000 \cdot 2 = 512000
\]

Now we can assess when the population reaches at least 500,000. From our calculations, we see that:

- At 24 days, the population is 256,000 (not enough).
- At 27 days, the population reaches 512,000 (which exceeds 500,000).

Thus, the minimum number of days required for the population to reach at least 500,000 is:

\[
\text{27 days}
\]