A house was valued at $110,000 in the year 1987. The value appreciated to $165,000 by the year 2004.
A) What was the annual growth rate between 1987 and 2004?
r
=
Round the growth rate to 4 decimal places.
B) What is the correct answer to part A written in percentage form?
r
=
%.
C) Assume that the house value continues to grow by the same percentage. What will the value equal in the year 2009 ?
value = $
Round to the nearest thousand dollars.
To find the annual growth rate between 1987 and 2004, we can use the formula:
Growth rate = (final value - initial value) / initial value
Growth rate = (165,000 - 110,000) / 110,000
Growth rate ≈ 0.5
Rounding to 4 decimal places:
Growth rate ≈ 0.4545
To find the growth rate in percentage form, we multiply by 100:
Growth rate ≈ 45.45%
To find the value in the year 2009, we need to use the growth rate to calculate the value after 5 years from 2004:
Value in 2009 = (1 + growth rate)^5 * Value in 2004
Value in 2009 ≈ (1 + 0.4545)^5 * 165,000
Value in 2009 ≈ 2.8326 * 165,000
Value in 2009 ≈ 466,041
Rounding to the nearest thousand dollars:
Value in 2009 ≈ $466,000
A) To find the annual growth rate between 1987 and 2004, we can use the formula:
r = (V2/V1)^(1/t) - 1
Where:
r = annual growth rate
V1 = initial value ($110,000 in 1987)
V2 = final value ($165,000 in 2004)
t = time in years (2004 - 1987 = 17 years)
Substituting the values into the formula, we have:
r = (165,000/110,000)^(1/17) - 1
Calculating this:
r = 1.0504 - 1
r ≈ 0.0504
Therefore, the annual growth rate between 1987 and 2004 is approximately 0.0504.
B) To express this growth rate in percentage form, we simply multiply it by 100:
r ≈ 0.0504 * 100
r ≈ 5.04%
Therefore, the correct answer to part A written in percentage form is approximately 5.04%.
C) Assuming the house value continues to grow by the same percentage, we need to find the value in the year 2009. Starting from the final value in 2004 ($165,000), we can calculate the growth over 5 years (2009 - 2004 = 5 years).
To calculate this, we can use the formula:
value = V2 * (1 + r)^t
Where:
value = final value in 2009 (to be determined)
V2 = final value in 2004 ($165,000)
r = growth rate (0.0504 from part A)
t = time in years (5 years)
Substituting the values into the formula, we have:
value = 165,000 * (1 + 0.0504)^5
Calculating this:
value ≈ 165,000 * 1.27416
value ≈ $210,204
Therefore, the value of the house in the year 2009, assuming the same growth rate, would be approximately $210,204 (rounded to the nearest thousand dollars).