To determine the tripling-time for the population of deer, we need to find the value of [tex]\( t \)[/tex] such that the population triples from its initial value. Let's go through the steps to solve this problem:
1. Define the initial conditions and the population growth formula:
- Initial population ([tex]\( P(0) \)[/tex]) = 988
- Growth formula: [tex]\( P(t) = 988 \times 3^{\frac{t}{2}} \)[/tex]
2. Define the condition for tripling:
We need to find the time [tex]\( t \)[/tex] when the population has tripled:
[tex]\[
3 \times \text{initial population} = \text{population at time } t
\][/tex]
This translates to:
[tex]\[
3 \times 988 = 988 \times 3^{\frac{t}{2}}
\][/tex]
3. Set up the equation and solve for [tex]\( t \)[/tex]:
First, cancel out the common factor of 988 from both sides of the equation:
[tex]\[
3 = 3^{\frac{t}{2}}
\][/tex]
4. Compare the exponents:
Because the bases are the same, the exponents must be equal:
[tex]\[
1 = \frac{t}{2}
\][/tex]
5. Isolate [tex]\( t \)[/tex]:
To find [tex]\( t \)[/tex], simply solve the equation for [tex]\( t \)[/tex]:
[tex]\[
t = 2
\][/tex]
So, the tripling-time for this population of deer is approximately [tex]\( \boxed{2} \)[/tex] years.
