In the year 1985, a house was valued at $112,000. By the year 2005, the value had appreciated exponentially to $140,000.
-What was the annual growth rate between 1985 and 2005?
-Assume that the value continued to grow by the same percentage. What was the value of the house in the year 2010?
v = 112,000 e^kt where t is years after 1985
2005 -1985 = 20
v(20) = 112,000 e^20 k = 140,000
so
e^20 k = 140/112
ln e^20 k = 20 k = ln (140/112)
k = .05 ln 1.25
solve for k
140 = 112 (1 + g)^20
log(140 / 112) = 20 log(1 + g)
1 + g = 10^{[log(140 / 112)] / 20}
2010 value = 112000 (1 + g)^25
put in t = 25 for value at 2010
for annual rate
that is dv/dt / v = k
To calculate the annual growth rate between 1985 and 2005, we can use the formula for exponential growth:
Growth Rate = (Final Value / Initial Value) ^ (1 / Number of Years) - 1
Let's plug in the given values:
Initial Value (1985) = $112,000
Final Value (2005) = $140,000
Number of Years = 2005 - 1985 = 20 years
Growth Rate = (140,000 / 112,000) ^ (1 / 20) - 1
Now we can calculate the growth rate:
Growth Rate ≈ (1.25) ^ (0.05) - 1
Growth Rate ≈ 0.0607 or 6.07%
So, the annual growth rate between 1985 and 2005 is approximately 6.07%.
To find the value of the house in 2010, we need to use the growth rate to calculate how much it increased from 2005 to 2010.
Number of Years = 2010 - 2005 = 5 years
Value in 2010 = Final Value (2005) * (1 + Growth Rate) ^ Number of Years
Plugging in the values:
Value in 2010 ≈ 140,000 * (1 + 0.0607) ^ 5
Calculating:
Value in 2010 ≈ 140,000 * (1.0607) ^ 5
Value in 2010 ≈ 140,000 * 1.3474
Value in 2010 ≈ $188,636
So, the value of the house in the year 2010 would be approximately $188,636, assuming the same growth rate continued.