A city of population 2.0 million is expected to experience a 15% decrease in population every 10 years.
a. What is the 10-year decay factor? What is the yearly decay factor? The yearly decay rate?
b. Use part (a) to create an exponential population model g(t) that gives the population (in millions) after t years.
c. Create an exponential population model h(t) that gives the population (in millions) after t years, assuming a 1.71% continuous yearly decrease.
d. Compare the populations predicted by the two functions after 20 years. What can you conclude?
My work:
decay factor: 1.15
yearly decay: 1.15^12= 5.3503
The yearly decay rate:35.03%
a. The 10-year decay factor is 1 - 0.15 = 0.85. The yearly decay factor can be calculated by taking the 10-year decay factor to the power of 1/10, which gives 0.85^(1/10) ≈ 0.9848. The yearly decay rate can be calculated by subtracting the yearly decay factor from 1 and converting it to a percentage, which gives (1 - 0.9848) * 100 ≈ 1.52%.
b. The exponential population model g(t) can be calculated using the formula: g(t) = 2.0 million * (0.9848)^t, where t is the number of years.
c. The exponential population model h(t) can be calculated using the formula: h(t) = 2.0 million * e^(-0.0171*t), where t is the number of years and e is the base of the natural logarithm (approximately 2.71828).
d. To compare the populations predicted by the two functions after 20 years, we can substitute t = 20 into the formulas for g(t) and h(t) and calculate the respective populations. g(20) ≈ 2.0 million * (0.9848)^20 ≈ 1.527 million. h(20) ≈ 2.0 million * e^(-0.0171*20) ≈ 1.590 million.
From these calculations, we can conclude that the population predicted by the exponential population model h(t) is higher than the population predicted by the exponential population model g(t) after 20 years.
a. To calculate the 10-year decay factor, we need to subtract 15% (or 0.15) from 1, which gives us 1 - 0.15 = 0.85. This means that after 10 years, the population will be 85% of its original value.
To find the yearly decay factor, we need to take the 10-year decay factor and find its 12th root. In this case, the 12th root of 0.85 is approximately 0.9534. This represents the decay factor over one year.
To find the yearly decay rate, we subtract the yearly decay factor from 1 and multiply by 100 to get a percentage. So, (1 - 0.9534) * 100 = 4.66%. Therefore, the yearly decay rate is 4.66%.
b. The exponential population model g(t) can be written as:
g(t) = 2.0 * (0.85^t)
Here, t represents the number of 10-year intervals. Since the initial population is 2.0 million, we multiply it by the decay factor raised to the power of t.
c. The continuous exponential decay formula is given by:
h(t) = 2.0 * e^(-0.0171t)
Here, t represents the number of years, and e is Euler's number (approximately equal to 2.71828).
We multiply the initial population (2.0 million) by the exponential decay function, which has a decay rate of 1.71% per year.
d. To compare the populations predicted by the two functions after 20 years, we can substitute t = 20 into both models.
For g(t): g(20) = 2.0 * (0.85^20) ≈ 0.473
For h(t): h(20) = 2.0 * e^(-0.0171 * 20) ≈ 1.472
Comparing the two values, we can conclude that the population predicted by the continuous exponential decay model (h(t)) is significantly higher than the population predicted by the discrete decay model (g(t)) after 20 years.