A population of wolves in a country is represented by the equation p(t)=80(0.98)^t, where t is the number of years since 1998. Predict the number of wolves in the population in the year 2008.
how many years will it take for the population of wolves to reach 500?
To predict the number of wolves in the population in the year 2008, we need to substitute t = 2008 - 1998 = 10 into the equation p(t).
p(10) = 80 * (0.98)^10
Calculating this expression will give us the predicted number of wolves in the population in the year 2008.
To determine how many years it will take for the population of wolves to reach 500, we need to set p(t) equal to 500 and solve for t.
Given: p(t) = 80 * (0.98)^t
Using the equation p(t) = 500, we can substitute 500 for p(t).
500 = 80 * (0.98)^t
Now we need to solve for t.
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.