To predict the number of wolves in the population in 2008, we need to substitute the value of t as the number of years since 1998. In 2008, the number of years since 1998 is 2008 - 1998 = 10.
Therefore, we substitute t = 10 into the equation p(t) = 80(0.98)^t:
p(10) = 80(0.98)^10
To calculate this, we raise 0.98 to the power of 10, then multiply the result by 80:
p(10) = 80(0.98)^10 ≈ 80(0.817) ≈ 65.36
So, in the year 2008, the population of wolves is predicted to be approximately 65.36.
Now, let's find out how many years it will take for the population of wolves to reach 500.
We need to solve the equation p(t) = 500 for t:
500 = 80(0.98)^t
First, divide both sides of the equation by 80 to isolate the exponential term:
(0.98)^t = 500/80
(0.98)^t = 6.25
Next, take the logarithm of both sides of the equation to solve for t. Since there is no specific base mentioned in the equation, we can use the natural logarithm (ln) or common logarithm (log):
ln[(0.98)^t] = ln(6.25)
t * ln(0.98) = ln(6.25)
Solve for t by dividing both sides of the equation by ln(0.98):
t = ln(6.25)/ln(0.98)
Using a calculator, we can find the approximate value of t:
t ≈ 33.8
Therefore, it will take approximately 33.8 years for the population of wolves to reach 500.