On Jan 1, 2010, Chessville has a population of 50,000 people. Chessville then enters a period of population growth. Its population increases 7% each year. On the same day, Checkersville has a population of 70,000 people. Checkersville starts to experience a population decline. its population decrease 4% each year. During what year will the population Chessville first exceed that of Checkersville? Show work and explain steps.
My answer is 3.11.
Chessville : 50000(1.07)^n
Checkervill: 70000(.96)^n
when is 50000(1.07)^n = 70000(.96)6n
5(1.07)^n = 7(.96)^n
log both sides and use rules of logs
log5 + nlog1.07) = log7 + nlog.96
nlog1.07 - nlog.96 = log7 - log5
n(log1.07 -log.96) = log7 - log5
n = ( log7 - log5)/(log1.07 -log.96)
= 3.101
in about 3 years from 2010 or in the year 2013
check:
50000(1.07)^3 = appr 61252
70000(.96)^3 = 61931
seems plausable, had I used the decimals it would be closer
Thanks!
To find out the year when the population of Chessville will first exceed that of Checkersville, we need to calculate the population of each city for each year until Chessville surpasses Checkersville. Here's how we can do it:
1) Start with the initial populations of both cities:
- Chessville: 50,000
- Checkersville: 70,000
2) Determine the annual population growth/decline rate for each city:
- Chessville: 7% increase each year
- Checkersville: 4% decrease each year
3) Calculate the populations for subsequent years until Chessville's population surpasses Checkersville's.
Year 1:
- Chessville: 50,000 + (7% of 50,000)
= 50,000 + 3,500
= 53,500
- Checkersville: 70,000 - (4% of 70,000)
= 70,000 - 2,800
= 67,200
Year 2:
- Chessville: 53,500 + (7% of 53,500)
= 53,500 + 3,745
= 57,245
- Checkersville: 67,200 - (4% of 67,200)
= 67,200 - 2,688
= 64,512
Year 3:
- Chessville: 57,245 + (7% of 57,245)
= 57,245 + 4,010
= 61,255
- Checkersville: 64,512 - (4% of 64,512)
= 64,512 - 2,580.48
= 61,931.52
4) We can see that in the third year, Chessville's population (61,255) exceeds Checkersville's population (61,931.52).
Therefore, the correct answer is that in the third year, the population of Chessville first exceeds that of Checkersville. The answer you provided, "3.11", seems to imply that it took around 3.11 years for Chessville to surpass Checkersville, which is not an accurate representation of the year.