Question

population math

1. write the equation for the rule of 70. explain its use.
2. complete the following calculations. show all work.

(a) a city has a population of 50,000 in 2012. if the population of the city grows at an annual rate of 2%, in what year will the population reach 100,000? in what year will the population reach 200,000?
(b) a countrys population was 12 million in 1992 and 24 million in 2024. if the population grew at a constant rate, what was its rate of growth as a percentage?
(c) in a particular year a population has the following characteristics: the crude birth rate is 45, the crude death rate is 20, the immigration rate is 1%, and the emigration rate is 0.5%. what is the percent rate of growth for the year?

Explanation:

Step1: Use the rule of 70 formula

The rule of 70 formula is \(T=\frac{70}{r}\), where \(T\) is the time (in years) it takes for a quantity to double and \(r\) is the growth rate (as a percentage). It is used to quickly estimate the time it takes for a population (or any quantity growing exponentially) to double.

Step2: Solve part (a)

• For population to reach \(100000\) (double of \(50000\)):

Using the rule of 70, \(T=\frac{70}{2}=35\) years. So the year is \(2012 + 35=2047\).

• For population to reach \(200000\) (double of \(100000\)):

Since it takes another \(35\) years to double from \(100000\) to \(200000\) (using the same growth rate), the year is \(2047+35 = 2082\).

Step3: Solve part (b)

The time period \(t=2024 - 1992=32\) years. The population doubles (\(P_0 = 12\) million, \(P = 24\) million). Using the rule of 70, \(T=\frac{70}{r}\), and since \(T = 32\), then \(r=\frac{70}{32}\approx2.19\%\)

Step4: Solve part (c)

The natural - growth rate (from birth and death rates) is \(r_{natural}=(45 - 20)\div1000=0.025\) (since crude birth and death rates are per 1000). The net - migration rate \(r_{migration}=(1 - 0.5)\%=0.005\). The total growth rate \(r=r_{natural}+r_{migration}=0.025 + 0.005=0.03\) or \(3\%\)

Answer:

1. The rule of 70 formula is \(T = \frac{70}{r}\), used to estimate doubling - time of a growing quantity.

2.

• (a) Population reaches \(100000\) in \(2047\) and \(200000\) in \(2082\).
• (b) Growth rate is approximately \(2.19\%\).
• (c) Percent rate of growth is \(3\%\).