Gce Model Questions Vol1 2018 Economics Question 15

A country's population follows the logistic growth model \( \frac{dP}{dt}= rP\left(1-\frac{P}{K}\right) \) with intrinsic growth rate \( r = 0.04 \) per year, carrying capacity \( K = 50 \) million and initial population \( P_{0}=20 \) million. The current age‑structure is 0‑14: 30 %, 15‑64: 60 %, 65+: 10 %. As the total population approaches \( K \), the proportion of 0‑14 declines linearly to 20 % at \( P=K \) and the proportion of 65+ rises linearly to 15 % at \( P=K \); the 15‑64 proportion adjusts accordingly. a) Estimate the population after 10 years. b) Compute the dependency ratio (dependents 0‑14 and 65+ versus 15‑64) after 10 years (as a percentage, one decimal place). c) To keep the dependency ratio below 55 %, what is the maximum population that can be reached, expressed as a percentage of the carrying capacity \( K \) (rounded to the nearest whole percent)?