Given:
- The number of individuals doubles every 10 days.
- Current number of individuals = 10,000
- Total available resources = 80,000
We need to determine the number of days it will take for the number of individuals to reach or exceed 80,000.
The number of individuals can be modeled by the exponential growth formula:
\[ N(t) = N_0 \cdot 2^{\frac{t}{T}} \]
where:
- \( N(t) \) is the number of individuals at time \( t \).
- \( N_0 \) is the initial number of individuals (10,000).
- \( T \) is the doubling time (10 days).
- \( t \) is the time in days.
We need to find \( t \) such that \( N(t) = 80,000 \):
\[ 80,000 = 10,000 \cdot 2^{\frac{t}{10}} \]
Dividing both sides by 10,000:
\[ 8 = 2^{\frac{t}{10}} \]
Taking the logarithm base 2 of both sides:
\[ \log_2(8) = \frac{t}{10} \]
Since \( 8 = 2^3 \):
\[ 3 = \frac{t}{10} \]
Solving for \( t \):
\[ t = 3 \cdot 10 \]
\[ t = 30 \]
Therefore, the resources will be exhausted in 30 days.
Correct Answer:A. 30
