A house was valued at $110,000 in the year 1985. The value appreciated to $145,000 by the year 2005.
-What was the annual growth rate between 1985 and 2005? Assume that the house value continues to grow by the same percentage.
-What did the value equal in the year 2010?
let 1985 <---> t = 0
then 2005 --> t = 20
2010 -----> t = 25
let the rate of appreciation be i (just like interest rate with money)
110000(1+i)^20 = 145000
(1+i)^20 = 1.3181818..
take 20th root
1+i = 1.013909
i = .0139 or appr 1.39 % (that's lousy)
value in 2010
= 110000(1.013909)^25
= $155,368
What do you have too do with the 20 root
Well, well, well, it looks like we have a question about the value of a house. Let's try not to get too house-torical here!
To calculate the annual growth rate between 1985 and 2005, we need to find the total growth over that period. The house value increased from $110,000 to $145,000, which means it grew by $145,000 - $110,000 = $35,000.
Now, let's calculate the number of years between 1985 and 2005. It's 2005 - 1985 = 20 years.
To find the annual growth rate, we divide the total growth ($35,000) by the number of years (20): $35,000 / 20 = $1,750.
So, the annual growth rate between 1985 and 2005 is $1,750. This means that the house value increased by $1,750 per year during that period. That's one fancy appreciation rate!
Now, to calculate the value in the year 2010, we'll need to continue growing it by the same percentage. From 1985 to 2005, the house value increased by $35,000. So, from 2005 to 2010 (a period of 5 years), it would increase by the same amount of $35,000.
Adding $35,000 to the value in 2005 ($145,000) gives us $145,000 + $35,000 = $180,000. That's what the house value would be in the year 2010.
So, in the year 2010, the value of the house would be $180,000. I hope the house is as funny as these math calculations!
To find the annual growth rate between 1985 and 2005, we can use the compound interest formula:
\[ A = P \times \left(1 + r\right)^n \]
Where:
- A is the future value of the house ($145,000 in 2005)
- P is the initial value of the house ($110,000 in 1985)
- r is the annual growth rate (to be determined)
- n is the number of years (2005 - 1985 = 20)
Plugging in the given values, we have:
\[ 145,000 = 110,000 \times \left(1 + r\right)^{20} \]
Now we can solve for r:
\[ \left(1 + r\right)^{20} = \frac{145,000}{110,000} \]
\[ \left(1 + r\right)^{20} = 1.31818 \]
Taking the 20th root of both sides to isolate (1 + r), we get:
\[ 1 + r = \sqrt[20]{1.31818} \approx 1.0285 \]
Subtracting 1 from both sides to find r, we have:
\[ r \approx 0.0285 \]
Therefore, the annual growth rate between 1985 and 2005 is approximately 2.85%.
To find the value of the house in the year 2010, we need to calculate the future value using the same growth rate. Since 2005 to 2010 is a span of 5 years, we can use the compound interest formula:
\[ A = P \times \left(1 + r\right)^n \]
Where:
- A is the future value of the house in the year 2010
- P is the value of the house in 2005 ($145,000)
- r is the annual growth rate (2.85%)
- n is the number of years (2010 - 2005 = 5)
Plugging in the given values, we have:
\[ A = 145,000 \times \left(1 + 0.0285\right)^5 \]
Calculating this expression:
\[ A \approx 145,000 \times 1.15073 \]
\[ A \approx 166,763 \]
Therefore, the value of the house in the year 2010 is approximately $166,763.
To find the annual growth rate between 1985 and 2005, we can use the formula for compound interest:
Annual Growth Rate = (Ending Value / Beginning Value) ^ (1 / Number of Years) - 1
In this case, the Beginning Value is $110,000 (1985), the Ending Value is $145,000 (2005), and the Number of Years is 2005 - 1985 = 20.
Putting these values into the formula, we get:
Annual Growth Rate = ($145,000 / $110,000) ^ (1 / 20) - 1
Simplifying this expression, we get:
Annual Growth Rate ≈ 0.0327 or 3.27%
Therefore, the annual growth rate between 1985 and 2005 is approximately 3.27%.
To find the value in the year 2010, we can use the formula for compound interest:
Value in Year 2010 = Beginning Value * (1 + Annual Growth Rate) ^ Number of Years
In this case, the Beginning Value is $145,000 (2005), the Annual Growth Rate is 3.27%, and the Number of Years is 2010 - 2005 = 5.
Plugging in these values into the formula, we get:
Value in Year 2010 = $145,000 * (1 + 0.0327) ^ 5
Using a calculator to evaluate this expression, we find:
Value in Year 2010 ≈ $167,689.82
Therefore, the value of the house in the year 2010 is approximately $167,689.82.