Answer:
a.
The growth rate (k) can be calculated using the exponential growth formula: P = P0 * e^(kt), where P0 is the initial population, t is the time in years, and e is the natural logarithm base.
From the information provided, the initial population of elves in 2000 was 23, and in 2003, the population was 34. By substituting the values into the formula and solving for k, we get:
34 = 23 * e^(k * 3)
Taking the natural logarithm of both sides:
ln(34) = ln(23 * e^(k * 3))
Using the logarithmic identity:
ln(34) = ln(23) + ln(e^(k * 3))
Solving for k:
k = (ln(34) - ln(23)) / (3) = 0.175
Therefore, the growth rate (k) for the population of elves is 0.175.
b.
To find the population of elves in 2020, we can use the exponential growth formula:
P = P0 * e^(kt)
Substituting the initial population of elves in 2000, which was 23, and the growth rate (k), which we found to be 0.175, we get:
P = 23 * e^(0.175 * 20)
Calculating the exponential function:
P = 23 * e^3.5
Therefore, there will be approximately 51 elves in 2020.
c.
To find the year when there will be 543 elves, we can use the exponential growth formula and solve for t:
543 = 23 * e^(0.175 * t)
Dividing both sides by 23 and taking the natural logarithm of both sides:
ln(543/23) = 0.175 * t
Solving for t:
t = (ln(543/23)) / 0.175
Therefore, there will be 543 elves in the year 2027.
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