Final answer:
It will take approximately 8 years for a home bought for $200,000 to appreciate at a rate of 5.4% per year to be worth at least $300,000. The calculation involves the formula for exponential growth and taking the natural logarithm to solve for time.
Step-by-step explanation:
To determine how long it will take for a home bought for $200,000 to appreciate to at least $300,000 at a rate of 5.4% per year, we use the formula for exponential growth. The formula is P = P_0 (1 + r)^t, where P is the future value, P_0 is the initial value, r is the rate of appreciation, and t is the time in years.
First, we substitute the values:
- P = $300,000 (the future value we want to reach)
- P_0 = $200,000 (the initial value of the home)
- r = 0.054 (5.4% written as a decimal)
Our equation is: $300,000 = $200,000 (1 + 0.054)^t
We divide both sides by $200,000 to isolate the exponential term:
1.5 = (1 + 0.054)^t
We then apply the natural logarithm to both sides to solve for t:
ln(1.5) = t * ln(1.054)
Divide by ln(1.054) to get t:
t = ln(1.5) / ln(1.054)
Using a calculator, we find t:
t ≈ 8.033
It will take approximately 8 years for the home to appreciate to at least $300,000 at a rate of 5.4% per year.