Question

the population of moscow, russia, can be modeled by the function
\\( p(t) = 9870(1.019)^t \\)
where \\( p(t) \\) is in thousands and \\( t \\) is the number of years since 2001.
(a) using this model, estimate the population in the year 2018.
(b) if this growth rate continues, in what year will the population reach 15 million (15 million is 15,000 thousand)?
part 1 of 2
(a) using this model, predict the population in the year 2018. round to the nearest thousand.
the population will be approximately \\( \square \\) thousand people.
part 2 of 2
(b) if this growth rate continues, in what year will the population reach 15 million (15 million is 15,000 thousand)?
the population will reach 15 million in the year \\( \square \\).

Explanation:

Part (a)

Step1: Calculate t for 2018

t = 2018 - 2001 = 17

Step2: Substitute t into P(t)

$P(17) = 9870(1.019)^{17}$
Calculate $(1.019)^{17}$: Using a calculator, $(1.019)^{17} \approx 1.394$
Then $P(17) \approx 9870 \times 1.394 \approx 13760$ (rounded to nearest thousand)

Step1: Set P(t) = 15000

$15000 = 9870(1.019)^{t}$

Step2: Solve for t

Divide both sides by 9870: $\frac{15000}{9870} = (1.019)^{t}$
Simplify: $\approx 1.519 = (1.019)^{t}$
Take natural log: $\ln(1.519) = t\ln(1.019)$
Solve for t: $t = \frac{\ln(1.519)}{\ln(1.019)} \approx \frac{0.429}{0.0188} \approx 22.8$

Step3: Find the year

Year = 2001 + 23 (since t ≈22.8, round up) = 2024

Answer:

13760

Part (b)