Question

Your grandfather likes to tell the story about how he started with 50 head of cattle on his ranch and grew the ranch to 1,000 head of cattle. He said "My plan was simple: grow the number of head of cattle at a rate of 10% per year."

Your grandfather reached his goal before he retired. How long did it take him?
A) 18.63 years
B) 31.43 years
C) 25.22 years
D) 47.65 years

Answer 1

31.43 years

Step-by-step explanation:

The number of years can be calculated using growth rate formula

yf = yi ( 1 + r) ^ t where yf is the final population, yi is the initial population, r is the rate in % and t is the number of years

yi = 50, yf = 1000, r = 10% and t = ?

substitute the values into the formula

1000 = 50 ( 1 + (10/100)) ^ t

divide both side by 50

1000/ 50 = 1000/50 ( 1.1) ^t

20 = (1.1) ^t

take log of both side

log 20 = t log 1.1 ( remember log a^b = b log a)

divide both side by log 1.1

log 20 / log 1.1 = t

t = 31.43 years