A certain species of deer is to be introduced into a forest, and wildlife experts estimate the population will grow to [tex] P(t) = 988 \cdot 3^{\frac{t}{2}} [/tex], where [tex] t [/tex] represents the number of years from the time of introduction.

Step 1 of 2: What is the tripling time for this population of deer?

Answer in years: _______



Answer :

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.