Question

13 the function $y = a(1.20)^t$ models the value of an investment after $t$ years. based on the function, what is the approximate monthly interest rate?
a 8.9%
c 1.5%
b 8.3%
d 1.0%

○ d
○ a
○ c
○ b

Explanation:

Step1: Understand the annual growth factor

The function \( y = a(1.20)^t \) has an annual growth factor of \( 1.20 \), meaning the annual growth rate is \( 20\% \) (since \( 1 + r = 1.20 \), so \( r = 0.20 \) or \( 20\% \) per year).

Step2: Convert annual rate to monthly rate

Let the monthly growth factor be \( 1 + i \), where \( i \) is the monthly interest rate (in decimal). There are 12 months in a year, so over 12 months, the growth factor should be equal to the annual growth factor. So we set up the equation:

$$ (1 + i)^{12} = 1.20 $$

To solve for \( i \), take the 12th root of both sides:

$$ 1 + i = 1.20^{\frac{1}{12}} $$

Calculate \( 1.20^{\frac{1}{12}} \). Using a calculator, \( 1.20^{\frac{1}{12}} \approx 1.0153 \). So \( i \approx 1.0153 - 1 = 0.0153 \), or \( 1.53\% \), which is approximately \( 1.5\% \).

Answer:

C. 1.5%