These questions are your test part 2 for this lesson. Please hand write or complete on a separate document and upload that file or photo to complete this test part 2.
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. 13³ = 2197
3. Solve ln2+lnx=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.
Apologies, but I'm unable to generate a handwritten or separate document. However, I can provide you with the answers to the questions.
1. To solve this problem, we can use the formula for exponential growth:
P = P0 * (1 + r)^t
Where:
P0 = initial population
P = final population
r = growth rate
t = time (in years)
In this case, P0 = 168,979, r = 1% = 0.01, and P = 180,000. We need to find the value of t. Rearranging the formula:
P/P0 = (1 + r)^t
180,000/168,979 = (1 + 0.01)^t
Divide both sides by 168,979:
1.065 = 1.01^t
Take the logarithm (base 10) of both sides:
log(1.065) = log(1.01^t)
Using the property of logarithms (log(a^b) = b*log(a)):
log(1.065) = t*log(1.01)
Divide both sides by log(1.01):
t = log(1.065) / log(1.01)
Using a calculator, we find:
t ≈ 6.677
Therefore, it will take approximately 6.677 years for the population of Tallahassee to reach 180,000.
2. The equation 13³ = 2197 can be written in logarithmic form as:
log base 13 of 2197 = 3
3. To solve the equation ln(2) + ln(x) = 3, we can combine the logarithms using the property of logarithms:
ln(2x) = 3
Next, we can exponentiate both sides using the natural exponential function:
e^(ln(2x)) = e^3
Simplifying, we get:
2x = e^3
Now, solve for x by dividing both sides by 2:
x = (e^3)/2
Using a calculator, we get:
x ≈ 5.565
Rounding to the nearest thousandth, x ≈ 5.565.
4. a. The exponential function to model the squirrel population in terms of the number of years from now is:
P(t) = 3310 * (1 + 0.04)^t
b. In this model, P(t) represents the squirrel population after t years, 3310 is the initial population, 0.04 represents the growth rate (4%), and t is the number of years from now.
c. To predict the number of squirrels after five years, we substitute t = 5 into the exponential function:
P(5) = 3310 * (1 + 0.04)^5
P(5) = 3310 * (1.04)^5
Using a calculator, we find:
P(5) ≈ 3872.37
Therefore, we predict that there will be approximately 3872 squirrels in the region after five years.