What was the percentage increase rate?

Question

According to the US Bureau of the Census, the world population in 1950 was 2,555,360,972. In 2003 it was 6,302,486,693. Based on these figures, assuming that the population increased by the same percentage each year from 1950 to 2003, what percent was that yearly increase? The corresponding figures for the US are populations of 152,271,000 in 1950 and 290,343,000 in 2003. This corresponds to an annual growth rate of what percent?

Hint: Enter a radical expression, or a decimal approximation with least two correctly rounded digits beyond the decimal point.

Answer 1

The most accurate growth rate formula is P = Po e^rt.

This is where:

P is the final population

Po is the initial population

e is a irrational number found on almost any scientific or graphing calculator and is about 2.72

r is the rate of growth (decimal representation of the % growth).

t is the change in time

To isolate for r:

P = Po e^rt <isolate the exponential function by dividing both sides by Po>

Po Po

P/Po = e^{rt < isolate rt by taking the natural log of both sides>}

ln(P/Po) = ln(e^rt )

ln(P/Po) = rt <isolate r by dividing by t>

ln(P/Po)/t = r or

r = ln(P/Po)/t

So from 1950 to 2003:

• The initial population is 2,555,360,972 (Po=2,555,360,972).

1. The final population is 6,302,486,693 (P=6,302,486,693).
2. r is unknown we will solve for it
3. The change in time is 2003-1950= 53 (t=53)

So substitute in the given values to solve for r as a decimal then multiply by 100 to find r as a %.

r[%] = 100ln(P/Po)/t = 100 ln (6,302,486,693/2,555,360,972)/53 ≅1.70%