A house was valued at $105,000 in the year 1990. The value appreciated to $145,000 by the year 2000.

Use the compound interest formula
S
=
P
(
1
+
r
)
t
to answer the following questions.

A) What was the annual growth rate between 1990 and 2000?
r
=  Round the growth rate to 4 decimal places.

We are given the compound interest formula:

\[
S = P(1 + r)^t
\]

Where:
- $ S $ is the final amount (the value in 2000, which is \$145,000),
- $ P $ is the initial amount (the value in 1990, which is \$105,000),
- $ r $ is the annual growth rate (which we need to find),
- $ t $ is the number of years (from 1990 to 2000, so $ t = 10 $).

We can substitute the given values into the formula and solve for $ r $:

\[
145,000 = 105,000 (1 + r)^{10}
\]

Now, let's solve this equation step by step.

First, divide both sides by 105,000:

\[
\frac{145,000}{105,000} = (1 + r)^{10}
\]

\[
\frac{145}{105} = (1 + r)^{10}
\]

\[
\frac{145}{105} \approx 1.38095
\]

Next, take the 10th root of both sides to solve for $ 1 + r $:

\[
(1 + r) = \left(1.38095\right)^{\frac{1}{10}}
\]

Finally, subtract 1 from both sides to find $ r $:

\[
r = \left(1.38095\right)^{\frac{1}{10}} - 1
\]

Let's calculate the value of $ r $.The annual growth rate $ r $ between 1990 and 2000 is approximately **0.0328**, or 3.28% when rounded to four decimal places.

Would you like further details or have any questions about this process?

Here are 5 related questions you might consider:
1. How would the growth rate change if the value in 2000 was \$160,000?
2. What is the house's estimated value in 2010 if the growth rate remains constant?
3. How does the compound interest formula apply in other real-world financial contexts?
4. Can we use this method to estimate inflation rates over the same period?
5. How would the growth rate differ if the time span was only 5 years?

**Tip:** When solving for growth rates, ensure that all values (especially time) are in consistent units to avoid errors.

A house was valued at $105,000 in the year 1990. The value appreciated to $145,000 by the year 2000. What was the annual growth rate between 1990 and 2000? Use the compound interest formula S = P(1 + r)^t and round the growth rate to 4 decimal places.

Learn how to calculate the annual growth rate of a house that appreciated from $105,000 in 1990 to $145,000 in 2000 using the compound interest formula. Step-by-step solution provided, including the compound growth formula and exponentiation. Suitable for Grades 10-12.

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Use the compound interest formula S

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