the population of Smallville in the year 1890 was 6250.assume the population increased at a rate of 2.75% per year.
predict when the population reached 50,000
I'm not suppose to use logs to find the answer :(
I need to use logs, sorry.
1.0275^n = 50,000/6250
thank you
To find when the population of Smallville reached 50,000 without using logarithms, we can calculate the population for each year until it reaches or exceeds 50,000.
Let's start with the given information:
Population in 1890: 6,250
Annual increase rate: 2.75%
To calculate the population for the next year (1891), we need to add 2.75% of the population in 1890 to the population in 1890:
Population in 1891 = 6,250 + (2.75/100) * 6,250
To calculate the population for the following year (1892), we repeat the process using the population in 1891:
Population in 1892 = Population in 1891 + (2.75/100) * Population in 1891
We continue this process until we find a year where the population reaches or exceeds 50,000.
Let's calculate the population for each year until we reach or exceed 50,000:
Population in 1890: 6,250
Population in 1891: 6,250 + (2.75/100) * 6,250
Population in 1892: Population in 1891 + (2.75/100) * Population in 1891
...
Keep repeating this process until you find a year where the population exceeds 50,000.
To predict when the population reached 50,000 in the year 1890, we can utilize a mathematical approach without the use of logarithms.
First, we need to determine the growth rate as a decimal. Given that the population increases by 2.75% per year, we can convert this to the decimal form by dividing by 100: 2.75 / 100 = 0.0275.
Next, we can use a recursive formula to determine the population of Smallville in each subsequent year. Let P(n) represent the population in year n, and P(0) = 6250 be the initial population in the year 1890.
The recursive formula can be written as:
P(n+1) = P(n) + 0.0275 * P(n)
Using the recursive formula, we can calculate the population each year until it reaches or exceeds 50,000. Since we start from 1890, we count the number of years until the population surpasses 50,000.
Let's calculate year by year until we reach a population greater than or equal to 50,000:
Year: 1890 Population: 6250
Year: 1891 Population: 6250 + (0.0275 * 6250) = 6414
Year: 1892 Population: 6414 + (0.0275 * 6414) = 6582
Year: 1893 Population: 6582 + (0.0275 * 6582) = 6754
Year: 1894 Population: 6754 + (0.0275 * 6754) = 6930
Year: 1895 Population: 6930 + (0.0275 * 6930) = 7110
Year: 1896 Population: 7110 + (0.0275 * 7110) = 7295.25
Since the population surpasses 50,000 between the years 1895 and 1896, we can conclude that the population reached 50,000 sometime during 1896.