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Leo is looking at two different savings plans. The first plan requires an initial deposit of [tex]$500 and grows at an annual interest rate of 2.5%. The second plan requires an initial deposit of $[/tex]400 and interest grows continuously at a rate of 2% per year.

Which two equations are part of the system representing the account balance of either plan, [tex]\( y \)[/tex], after [tex]\( x \)[/tex] years?

A. [tex]\( y = 400e^{0.02x} \)[/tex]
B. [tex]\( y = 500(1.025)^x \)[/tex]
C. [tex]\( y = 400e^{2x} \)[/tex]
D. [tex]\( y = 500(1.25)^x \)[/tex]
E. [tex]\( y = 500e^{0.025x} \)[/tex]
F. [tex]\( y = 400(1.02)^x \)[/tex]



Answer :

GinnyAnswer
To determine which two equations Leo wrote to represent the account balance of each plan, let's analyze the information given and identify the correct forms of the equations for each plan.

For the first savings plan:
- Initial deposit: \[tex]$500 - Annual interest rate: 2.5% This is a standard compound interest formula, given by \( y = P(1 + r)^x \), where: - \( P \) is the principal amount (\$[/tex]500)
- [tex]\( r \)[/tex] is the annual interest rate (2.5% or 0.025)
- [tex]\( x \)[/tex] is the number of years

So the equation for the first plan is:
[tex]\[ y = 500(1.025)^x \][/tex]

For the second savings plan:
- Initial deposit: \[tex]$400 - Continuous interest rate: 2% per year This uses the continuous compound interest formula, given by \( y = Pe^{rx} \), where: - \( P \) is the principal amount (\$[/tex]400)
- [tex]\( r \)[/tex] is the continuous interest rate (2% or 0.02)
- [tex]\( x \)[/tex] is the number of years

So the equation for the second plan is:
[tex]\[ y = 400e^{0.02x} \][/tex]

Comparing these to the given options, the correct answers are:
[tex]\[ y = 500(1.025)^x \][/tex]
[tex]\[ y = 400e^{0.02x} \][/tex]

These match with the following provided options:
- [tex]\( y = 500(1.025)^x \)[/tex]
- [tex]\( y = 400 e ^{0.02 x} \)[/tex]

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