Question

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?

Answer 1

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

Answer 2

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

Answer 3

put in t = 25 for value at 2010

for annual rate
that is dv/dt / v = k

Answer 4

grew by a factor of 140/112 = 1.25 in 20 years. So, the annual growth factor was 1.25^(1/20) = 1.0112 or 1.12%