The population of a town grows at the rate of 20% every 5 years. In how many years will it double itself (approximately)?
(a) 12
(b) 15
(c) 16
(d) 20
Answer (2)
The population of the town will take approximately 20 years to double, based on a growth rate of 20% every 5 years, using logarithmic calculations. The closest answer choice is therefore (d) 20. This conclusion is derived by applying the formula for exponential growth and logarithmic values to solve for time.
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To find out in how many years the town's population will double, we need to use the formula for exponential growth:
P ( t ) = P 0 × ( 1 + r ) t
where:
P ( t ) is the future population, P 0 is the initial population, r is the growth rate per period, t is the number of periods.
In this problem, we want the population to double, so P ( t ) = 2 P 0 . The growth rate is given as 20% every 5 years, so r = 0.2 . We need to find t such that:
2 P 0 = P 0 × ( 1 + 0.2 ) t /5
This simplifies to:
2 = ( 1.2 ) t /5
To solve for t , we take the logarithm of both sides:
lo g ( 2 ) = lo g (( 1.2 ) t /5 )
By using the logarithm power rule, we have:
lo g ( 2 ) = 5 t × lo g ( 1.2 )
Solving for t gives:
t = 5 × lo g ( 1.2 ) lo g ( 2 )
Using values: lo g ( 2 ) ≈ 0.3010 and lo g ( 1.2 ) ≈ 0.0792 , we calculate:
t = 5 × 0.0792 0.3010 ≈ 5 × 3.8 ≈ 19 years
So, the population will approximately double in 19 years. Therefore, the closest option in the given choices is (d) 20.
