you are depositing $1000 dollars in a savings account and are given the following options.
6.2% annual interest rate, compounded annually
6.1% annual interest rate, compounded quarterly
6.0% annual interest rate, compounded continuously
for each option, write a function that gives the balance as a function of the time (T) in years.
graph all three functions in the same viewing window on your calculator, describe the viewing window.
find the balances for the three options after 25,50,75, and 100 years. is the option that yields the greatest balance after 25 years the same option that yields the greatest balance after 50,75, and 100 years? explain
the effective yield of a savings plan is the percent increase in the balance after 1 year. find the efective yields for the three options listed above. how can the effective yield be used to decide which option is best?
IM SO LOST AND SO SWARMED WITH OTHER HOMEWORK. I have been in the hospital for two weeks after having surgery and today was my first day back. PLEASE HELP! :(
I'm sorry to hear about your situation, but I'm here to help you with your question. Let's break it down step by step.
To find the balance as a function of time for each option, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
where A is the balance, P is the principal (initial deposit), r is the annual interest rate, n is the number of times interest is compounded per year, and t is the time in years.
For the first option with a 6.2% annual interest rate, compounded annually, the function would be:
A1(T) = 1000(1 + 0.062/1)^(1T)
For the second option with a 6.1% annual interest rate, compounded quarterly, the function would be:
A2(T) = 1000(1 + 0.061/4)^(4T)
For the third option with a 6.0% annual interest rate, compounded continuously, the function would be:
A3(T) = 1000e^(0.06T)
Now let's graph these functions. To do this, you can use a graphing calculator or any online graphing tool. Set the viewing window in a way that includes the range of values you are interested in, such as 0 to 100 years for the x-axis and a reasonable range for the y-axis.
To find the balances after 25, 50, 75, and 100 years, you can simply plug in the respective values of T into each function. Calculate A1(25), A2(25), A3(25), A1(50), A2(50), A3(50), and so on. Compare the resulting balances.
The option that yields the greatest balance after 25 years may not necessarily be the same option that yields the greatest balance after 50, 75, and 100 years. This is because different compounding frequencies can affect the overall growth of the investment over time. By comparing the balances at different points in time, you can determine which option is best for your specific needs.
The effective yield of a savings plan is the percent increase in the balance after one year. To find the effective yield for each option, you can use the formula:
Effective Yield = (A - P)/P * 100
where A is the balance after one year and P is the principal (initial deposit). Calculate the effective yield for each option using the balance you obtained after one year, and compare them. The option with the highest effective yield can be considered the best option if you prioritize maximizing the return on your investment within a one-year timeframe.
Remember that these calculations are based on the provided interest rates and compounding frequencies. It's essential to consider other factors, such as fees, accessibility of funds, and your own financial goals, when choosing a savings account.