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%

Explanation:

Step1: Recall compound interest formula

The given function is \( y = a(1.20)^t \), where \( t \) is in years. For monthly compounding, we let \( t=\frac{m}{12} \) (where \( m \) is months) and rewrite the formula as \( y = a(1 + r)^{m} \), so for annual rate \( R = 0.20 \) (since \( 1.20=1 + 0.20 \)), we want to find the monthly rate \( r \) such that \( (1 + r)^{12}=1.20 \).

Step2: Solve for monthly rate \( r \)

Take the 12th root of both sides: \( 1 + r = 1.20^{\frac{1}{12}} \). Calculate \( 1.20^{\frac{1}{12}} \approx e^{\frac{\ln(1.20)}{12}} \) (or use a calculator). \( \ln(1.20)\approx0.1823 \), so \( \frac{0.1823}{12}\approx0.01519 \), then \( e^{0.01519}\approx1.0153 \). So \( r\approx0.0153 \), which is about \( 1.5\% \). Wait, no, wait: Wait, \( 1.20^{1/12} \): let's calculate directly. \( 1.20^{1/12}=\sqrt[12]{1.20} \). Let's compute step by step. \( 1.20^{1/12}\approx1.0153 \), so the monthly rate is about \( 1.5\% \)? Wait, no, wait, maybe I made a mistake. Wait, the annual growth factor is 1.20, so annual rate is 20%. To find monthly rate, we need to find \( r \) such that \( (1 + r)^{12}=1.20 \). So \( r = 1.20^{1/12}-1 \). Let's compute \( 1.20^{1/12} \). Let's use logarithm: \( \log_{10}(1.20)=0.07918 \), divide by 12: \( 0.07918/12\approx0.006598 \), then \( 10^{0.006598}\approx1.0153 \), so \( r\approx0.0153 \), which is 1.53%, so approximately 1.5%. Wait, but let's check again. Wait, maybe the function is \( y = a(1 + r)^t \), where \( t \) is years. So annual rate is 20%, so monthly rate: \( (1 + r)^{12}=1.20 \), so \( r = 1.20^{1/12}-1 \approx 1.0153 - 1 = 0.0153 \), so 1.53%, which is approximately 1.5%, so option C. Wait, but let's verify with another method. Let's compute \( 1.015^{12} \): \( 1.015^{12}=(1.015^6)^2 \). \( 1.015^6\approx1.0934 \), then squared is \( 1.0934^2\approx1.1956 \), which is close to 1.20. So yes, \( 1.015^{12}\approx1.20 \), so the monthly rate is approximately 1.5%.

Wait, but wait, maybe I messed up. Wait, the function is \( y = a(1.20)^t \), so \( t \) is in years. So after 1 year, the value is \( 1.20a \), so annual growth rate 20%. To find monthly rate, we need to find \( r \) such that after 12 months (1 year), the growth factor is 1.20. So \( (1 + r)^{12}=1.20 \), so \( r = 1.20^{1/12}-1 \approx 1.0153 - 1 = 0.0153 \), so 1.53%, which is approximately 1.5%, so option C.

Answer:

C. 1.5%