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.

Question

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