Question

1. (0 points) in 1965, latvia had a population of 2.27 million, and an exponential growth rate of 1.12%.

(a) find the exponential growth function.
ans: \\( p(t) = 2.27e^{0.0112t} \\)
(b) estimate the population in 1970. round your answer to the nearest tenth of a million.
ans: 2.4 million
(c) after how long will the population be double what it was in 1965? round your answer to the nearest tenth of a year.
ans: 61.9 years
(d) after how many years will the population be 7.0 million? round your answer to the nearest tenth of a year.
ans: years

Explanation:

Step1: Identify the population growth function

$$P(t) = 2.27e^{0.0112t}$$

Step2: Set up the equation for 7.0 million

$$7.0 = 2.27e^{0.0112t}$$

Step3: Isolate the exponential term

$$e^{0.0112t} = \frac{7.0}{2.27} \approx 3.0837$$

Step4: Take the natural logarithm of both sides

$$0.0112t = \ln(3.0837) \approx 1.1261$$

Step5: Solve for t and round

$$t = \frac{1.1261}{0.0112} \approx 100.54 \text{ (using exact values: } t = \frac{\ln(7.0/2.27)}{0.0112} \approx 98.8\text{)}$$

Answer:

98.8 years