The towns of Franklin and Chester each decided in the same year to begin posting their populations on signs by the main entrances into the towns. Each year, they update the signs.
Franklin had a population of 20,000 people in the first year that they posted the sign, and each year, the population increases by approximately 5%
Chester had a population of 25,000 people in the first year that they posted the sign, and each year, the population increases by 500 people.
In which year will Franklin's population first exceed Chester's population?
you want the geometric progression and the arithmetic progression to be equal after n years. That is,
20000*1.05^(n-1) = 25000+500(n-1)
n = 8.4
so, 8.4 years after they started posting populations, the signs will read the same.
Thus, after 9 years Franklin's population will be greater.
Well, let's do some calculations! We'll start with Franklin's population and see how it grows over the years.
In the first year, Franklin has 20,000 people. If the population increases by 5% each year, we can calculate the population for each subsequent year using the formula: population = previous year's population + (previous year's population * growth rate).
For example, in the second year, we have:
population = 20,000 + (20,000 * 0.05) = 20,000 + 1,000 = 21,000.
Now, let's see how Chester's population grows. In the first year, Chester has 25,000 people, and it increases by 500 people each year.
To find the year when Franklin's population exceeds Chester's population, we need to compare the populations year by year.
Here we go:
Year 1:
Franklin's population: 20,000
Chester's population: 25,000
Year 2:
Franklin's population: 21,000
Chester's population: 25,500
Year 3:
Franklin's population: 22,050
Chester's population: 26,000
Year 4:
Franklin's population: 23,152.5
Chester's population: 26,500
Year 5:
Franklin's population: 24,310.12
Chester's population: 27,000
Year 6:
Franklin's population: 25,525.63
Chester's population: 27,500
Year 7:
Franklin's population: 26,803.91
Chester's population: 28,000
Year 8:
Franklin's population: 28,149.10
Chester's population: 28,500
Year 9:
Franklin's population: 29,565.56
Chester's population: 29,000
Year 10:
Franklin's population: 31,057.84
Chester's population: 29,500
Drumroll, please! In Year 10, Franklin's population exceeds Chester's population for the first time!
So, it seems like it takes Franklin a solid ten years to overcome Chester. Hang in there, Franklin, slow and steady wins the race!
To determine in which year Franklin's population will first exceed Chester's population, we need to set up an equation and solve for the year.
Let's assume that the year in question is represented by 'n.'
For Franklin:
Population in year n = 20,000 * (1 + 0.05)^n
For Chester:
Population in year n = 25,000 + 500n
We can set up an equation to solve for 'n':
20,000 * (1 + 0.05)^n = 25,000 + 500n
Now, let's solve for 'n' by isolating it on one side of the equation:
(1 + 0.05)^n = (25,000 + 500n) / 20,000
Taking the natural logarithm (ln) of both sides:
ln[(1 + 0.05)^n] = ln[(25,000 + 500n) / 20,000]
Using the logarithmic property, we can move the exponent 'n' to the front:
n * ln(1 + 0.05) = ln[(25,000 + 500n) / 20,000]
Now, divide both sides of the equation by ln(1 + 0.05):
n = ln[(25,000 + 500n) / 20,000] / ln(1 + 0.05)
Using a calculator to evaluate both the numerator and denominator, we find:
n ≈ 9.36
Since the year must be a whole number, we can round up to the nearest integer:
n = 10
Therefore, Franklin's population will first exceed Chester's population in the 10th year.
To find the year when Franklin's population first exceeds Chester's population, we need to compare the population growth rates of both towns.
For Franklin, the population increases by approximately 5% each year. To calculate the population of Franklin in the next year, we can use the formula:
New population of Franklin = Current population of Franklin + (Current population of Franklin × 5%)
For Chester, the population increases by 500 people each year. To calculate the population of Chester in the next year, we can use the formula:
New population of Chester = Current population of Chester + 500
We can start by calculating the population of Franklin and Chester for the first year:
Franklin (Year 1) = 20,000 + (20,000 × 0.05) = 20,000 + 1,000 = 21,000
Chester (Year 1) = 25,000 + 500 = 25,500
Since Franklin's population is already smaller than Chester's population in the first year, we need to continue calculating their populations until Franklin's population exceeds Chester's population.
Franklin (Year 2) = 21,000 + (21,000 × 0.05) = 21,000 + 1,050 = 22,050
Chester (Year 2) = 25,500 + 500 = 26,000
Franklin (Year 3) = 22,050 + (22,050 × 0.05) = 22,050 + 1,102.5 = 23,152.5
Chester (Year 3) = 26,000 + 500 = 26,500
We can continue this process until Franklin's population exceeds Chester's population.
Franklin (Year 4) = 23,152.5 + (23,152.5 × 0.05) = 23,152.5 + 1,157.625 = 24,310.125
Chester (Year 4) = 26,500 + 500 = 27,000
From the calculations, we see that in Year 4, Franklin's population exceeds Chester's population. Therefore, the answer is Year 4.