Let's break down your questions step by step based on the given equation \( p(t) = 800(0.95)^t \).
1a. Is the population growing or decaying?
The population is decaying. This conclusion is based on the factor of 0.95 in the equation. Since 0.95 is less than 1, it indicates that the population decreases over time.
1b. At what rate is it increasing or decreasing?
The rate at which the population is decreasing can be determined using the formula for exponential decay. The general formula can be expressed in the form \( p(t) = p_0 \cdot e^{-rt} \). In your case, with \( p(t) = 800(0.95)^t \), we can determine the decay rate as follows:
- The decay factor is 0.95, which indicates a 5% decrease each time period \( t \) (since \( 1 - 0.95 = 0.05 \)).
- Therefore, the population decreases at a rate of 5% per time unit.
1c. What does the 800 represent?
The value 800 in the equation \( p(t) = 800(0.95)^t \) represents the initial population of coyotes when \( t = 0 \). In other words, there were 800 coyotes at the start.
2. How many coyotes are there after 12 weeks?
To find the number of coyotes after 12 weeks, substitute \( t = 12 \) into the equation:
\[ p(12) = 800(0.95)^{12} \]
Calculating \( (0.95)^{12} \):
\[ (0.95)^{12} \approx 0.5404 \]
Now substituting this back into the population function:
\[ p(12) = 800 \times 0.5404 \approx 432.32 \]
Rounding to the nearest whole number, we find:
\[ p(12) \approx 432 \]
Summary of Answers:
1a. The population is decaying. 1b. It is decreasing at a rate of 5% per time unit. 1c. The 800 represents the initial population of coyotes. 2. After 12 weeks, there are approximately 432 coyotes.
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