Question

A 200,000 house appreciates in value 10% a year. An equation used to represent this is y = 200000*1.10^t . Write an expression to represent the monthly appreciation.
Responses
A y = 200000 ( 1.008 ) ^t
B y = 200000 ( 1.008 ) ^12 t
C y = 200000 ( 3.138 )^ t
D y = 200000
t

Answer 1

To convert the annual appreciation of 10% to a monthly appreciation, we need to find the equivalent monthly growth rate.

The formula for the annual growth when compounded monthly can be derived from:

\[
(1 + r)^n = 1 + R
\]
where:
- \( R \) is the annual growth rate (0.10 for 10%),
- \( n \) is the number of compounding periods per year (12 for months),
- \( r \) is the monthly growth rate.

Setting it up, we get:

\[
(1 + r)^{12} = 1.10
\]

To find \( r \), we can solve for \( r \):

\[
1 + r = (1.10)^{1/12}
\]

Subtracting 1 gives us the monthly growth rate:

\[
r = (1.10)^{1/12} - 1
\]

Calculating this value, we find that:

\[
r \approx 0.00797414
\]

Thus, our monthly growth rate can be approximated as:

\[
1 + r \approx 1.008
\]

So, the expression for monthly appreciation, t being measured in months, becomes:

\[
y = 200000 \cdot (1.008)^t
\]

Therefore, the correct option representing the monthly appreciation is:

**A: \( y = 200000 \cdot (1.008)^t \)**