1. In 2023, the population of Tallahassee, Florida was 168, 979. Some
researchers believe that the population of Tallahassee will increase by a rate of
1% each year for the next 10 years.
If researchers are correct, how many years will it take for the population of
Tallahassee to reach 180,000?
Hint: You will need to use exponential growth formula and logarithms to solve.
2. Write the equation in logarithmic form. 13^3 = 2197
3. Solve In2+Inx=3. Round to the nearest thousandth, if necessary.
4. In the Olympic National Park, there are currently 3310 squirrels, and the
population is increasing at an annual rate of 4%.
a. Write an exponential function to model the squirrel population in terms of the
number of years from now.
b. Explain what each value in the model represents.
c. Predict the number of squirrels that will be in the region after five years. Show
your work.
1. We can solve this using the exponential growth formula:
Final Population = Initial Population * (1 + Growth Rate)^Number of Years
Let's use P as the final population, P0 as the initial population, r as the growth rate, and t as the number of years.
P = P0 * (1 + r)^t
We are given P0 = 168979, r = 0.01 (1% expressed as a decimal), and we need to find t when P = 180000.
180000 = 168979 * (1 + 0.01)^t
Divide both sides by 168979:
180000 / 168979 = 1.01^t
Using logarithms to solve for t, take the natural logarithm (ln) of both sides:
ln(180000 / 168979) = ln(1.01^t)
t * ln(1.01) = ln(180000 / 168979)
Divide both sides by ln(1.01):
t = ln(180000 / 168979) / ln(1.01)
Using a calculator, evaluate the right side: t ≈ 6.386
Therefore, it will take approximately 6.386 years for the population of Tallahassee to reach 180,000.
2. The equation 13^3 = 2197 can be written in logarithmic form as:
log base 13 (2197) = 3
So, log base 13 of 2197 equals 3.
3. We have the equation In2 + Inx = 3. To solve for x, we need to use logarithmic properties.
Combine the two logarithms:
In2x = 3
Rewrite this equation as an exponential form:
e^3 = 2x
Simplify the left side:
20.0855... = 2x
Take the natural logarithm (ln) of both sides:
ln(20.0855...) = ln(2x)
Solve for x:
x = ln(20.0855...) / ln(2)
Using a calculator, evaluate the right side: x ≈ 4.585
Therefore, x ≈ 4.585.
4. a. The exponential function to model the squirrel population in terms of the number of years from now is:
P = 3310 * (1 + 0.04)^t
b. In the model, P represents the squirrel population in the Olympic National Park after t years, 3310 represents the initial population, 0.04 represents the growth rate (4% expressed as a decimal), and t represents the number of years from now.
c. To predict the number of squirrels after five years, substitute t = 5 into the exponential function:
P = 3310 * (1 + 0.04)^5
P ≈ 3836.174
Therefore, the predicted number of squirrels in the region after five years is approximately 3836.