John, Sally, and Natalie would all like to save some money. John decides that it
would be best to save money in a jar in his closet every single month. He decides
to start with $300, and then save $100 each month. Sally has $6000 and decides
to put her money in the bank in an account that has a 7% interest rate that is
compounded annually. Natalie has $5000 and decides to put her money in the
bank in an account that has a 10% interest rate that is compounded continuously.
(1pt) What type of equation models John’s situation? _________________
(1pt) Write the model equation for John’s situation ___________________
(1pt) How much money will John have after 2 years? _________________
(1pt) How much money will John have after 10 years? ________________
(1pt) What type of exponential model is Sally’s situation? ______________
(2pt) Write the model equation for Sally’s situation ___________________
(1pt) How much money will Sally have after 2 years? _________________
(1pt) How much money will Sally have after 10 years? ________________
(1pt) What type of exponential model is Natalie’s situation? ____________
(2pt) Write the model equation for Natalie’s situation _________________
(1pt) How much money will Natalie have after 2 years? _______________
(1pt) How much money will Natalie have after 10 years? ______________
(2pts) Who will have the most money after 10 years?
***help please this is the correct one**
To answer these questions, we need to understand the different equations and models that represent John's, Sally's, and Natalie's situations.
John's situation is best represented by a linear equation because he saves a fixed amount of money every month. This equation is in the form of y = mx + b, where y represents the amount of money, x represents the number of months, m represents the monthly savings amount, and b represents the initial amount.
Given that John starts with $300 and saves $100 each month, the equation becomes:
y = 100x + 300.
Now let's answer the specific questions:
1. John will have after 2 years (24 months):
y = 100 * 24 + 300 = $2400.
2. John will have after 10 years (120 months):
y = 100 * 120 + 300 = $12,300.
Sally's situation can be modeled using the compound interest formula A = P(1 + r/n)^(nt), where A represents the final amount, P represents the principal amount (initial amount), r represents the interest rate (in decimal form), n represents the number of times the interest is compounded per year, and t represents the number of years.
In Sally's case, the interest is compounded annually, so n = 1. The equation becomes:
A = 6000(1 + 0.07/1)^(1*2), where 0.07 is the 7% interest rate and 2 is the number of years.
Now let's calculate the specific values:
1. Sally will have after 2 years:
A = 6000(1 + 0.07)^2 = $7122.
2. Sally will have after 10 years:
A = 6000(1 + 0.07)^10 = $12,861.96.
Natalie's situation is modeled using the continuous compound interest formula A = Pert, where A represents the final amount, P represents the principal amount, e represents Euler's number (approximately 2.71828), r represents the interest rate (in decimal form), and t represents the number of years.
In Natalie's case, the equation becomes:
A = 5000 * e^(0.10 * 2), where 0.10 is the 10% interest rate and 2 is the number of years.
Now let's calculate the specific values:
1. Natalie will have after 2 years:
A = 5000 * e^(0.10 * 2) = $6727.50.
2. Natalie will have after 10 years:
A = 5000 * e^(0.10 * 10) = $8187.10.
To compare who will have the most money after 10 years, we can simply compare the final amounts:
- John will have $12,300.
- Sally will have $12,861.96.
- Natalie will have $8,187.10.
Therefore, after 10 years, Sally will have the most money.