In the year 1985, a house was valued at $110,000. By the year 2005, the value had appreciated exponentially to $140,000.
1. What was the annual growth rate between 1985 and 2005?Round your answer to two decimal places.)
2. Assume that the value continued to grow by the same percentage. What was the value of the house in the year 2010? Round your answer to the nearest dollar.)
To find the annual growth rate between 1985 and 2005, we can use the formula for exponential growth:
Growth Rate = (Ending Value / Starting Value)^(1 / Number of Years) - 1
1. Calculating the annual growth rate between 1985 and 2005:
Starting Value (1985) = $110,000
Ending Value (2005) = $140,000
Number of Years = 2005 - 1985 = 20
Growth Rate = (140,000 / 110,000)^(1 / 20) - 1
Using a calculator or a spreadsheet, we can input this formula and find that the growth rate is approximately 0.0256 (or 2.56% when rounded to two decimal places).
2. To find the value of the house in the year 2010 assuming the same percentage growth rate:
Starting Value (2005) = $140,000
Number of Years = 2010 - 2005 = 5
Value in 2010 = Starting Value * (1 + Growth Rate)^Number of Years
Value in 2010 = $140,000 * (1 + 0.0256)^5
Using a calculator or a spreadsheet, we can calculate that the value of the house in 2010 is approximately $156,593.
let 1985 correspond with t = 0
then 2005 ---> t = 20
If we assume that the growth is continuous,
110000 e^20r = 140000
e^20r = 1.2727...
take ln of both sides, and use log rules
20r = ln (1.2727...) = .241162...
r = .0112058..
or appr 1.12%
so you would have
Value = 110000 e(.0112058..*t)
since 2010 ----> t = 25
sub in t = 25 and evaluate.
Let me know what you got.
If you wanted the rate just compounded annually, then solve
110000(1+r)^20 = 140000
you should get r = .01213