Question

select all the correct answers.
leo is looking at two different savings plans. the first plan requires an initial deposit of $500 and grows at an annual interest rate of 2.5%. the second plan requires an initial deposit of $400 and interest grows continuously at a rate of 2% per year.
leo wrote a system of equations to represent the account balance of either plan, y, after x years. which two equations are part of the system?

$y = 500(1.25)^x$
$y = 500e^{0.025x}$
$y = 500(1.025)^x$
$y = 400e^{2x}$
$y = 400e^{0.02x}$
$y = 400(1.02)^x$

Explanation:

Step1: Analyze First Savings Plan

The first plan has an initial deposit (principal) \( P = 500 \) and an annual interest rate \( r = 2.5\%=0.025 \). For annual compounding, the formula for the amount \( y \) after \( x \) years is \( y = P(1 + r)^x \). Substituting the values, we get \( y = 500(1 + 0.025)^x=500(1.025)^x \).

Step2: Analyze Second Savings Plan

The second plan has an initial deposit \( P = 400 \) and continuous compounding with rate \( r = 2\% = 0.02 \). The formula for continuous compounding is \( y = Pe^{rx} \). Substituting the values, we get \( y = 400e^{0.02x} \).

Answer:

\( y = 500(1.025)^x \), \( y = 400e^{0.02x} \)