Grace deposits $1000 in a mutual fund earning 9.25% annual interest, compounded monthly.
1. Write an exponential function that models this situation where y is the amount of Grace's investment and x is time in years.
2. Use your equation to complete the years and balance chart. (Round to the nearest cent.)
Years Balance
5
10
15
20
25
30
3.Use your data from problem 2 to graph your equation from problem 1. (Use an x-scale of 5 years and a y-scale of $1000.)
4. If Grace were to invest $1000 at age 25 and not withdraw any money until retirement at age 67, calculate the expected value of her investment. Show your calculations and consider the trend of your graph.
All help is appreciated! I suck at math and need help with it. I just don't understand it.
9.25% monthly
.095/12 = .0079167
so every month multiply by
1.0079167
y = 1,000 (1.0079167)^(12x)
now just activate your calculator :)
1. To model this situation, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the final amount (balance)
P = the initial principal (investment)
r = annual interest rate (as a decimal)
n = number of times interest is compounded per year
t = time in years
In this case, Grace's initial investment is $1000, the annual interest rate is 9.25% (or 0.0925 as a decimal), and interest is compounded monthly (n = 12). So the equation becomes:
y = 1000(1 + 0.0925/12)^(12x)
2. To complete the years and balance chart, we can plug the values of x (years) into the equation we just derived and calculate the corresponding balance (y).
Years Balance
5 y = 1000(1 + 0.0925/12)^(12*5)
10 y = 1000(1 + 0.0925/12)^(12*10)
15 y = 1000(1 + 0.0925/12)^(12*15)
20 y = 1000(1 + 0.0925/12)^(12*20)
25 y = 1000(1 + 0.0925/12)^(12*25)
30 y = 1000(1 + 0.0925/12)^(12*30)
Evaluate each equation to calculate the balance for each corresponding year.
3. To graph the equation, we can plot the years (x-axis) against the balance (y-axis). Using an x-scale of 5 years and a y-scale of $1000, we can plot the data points we obtained above and connect them to form a line.
4. To calculate the expected value of Grace's investment if she were to invest $1000 at age 25 and not withdraw any money until retirement at age 67, we need to calculate the balance at age 67 using the equation we derived earlier.
y = 1000(1 + 0.0925/12)^(12*(67-25))
Simplify the equation above and calculate the balance to find the expected value of Grace's investment at age 67.
1. To write an exponential function that models this situation, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the future value of the investment (balance)
P = the principal amount (initial deposit)
r = annual interest rate (as a decimal)
n = number of times interest is compounded per year
t = time in years
In this case, the principal amount (P) is $1000, the annual interest rate (r) is 9.25% or 0.0925, and the interest is compounded monthly (n = 12).
Therefore, the exponential function that models this situation is:
y = 1000(1 + 0.0925/12)^(12x)
2. To complete the years and balance chart, we can substitute the values of x in the exponential function and calculate the corresponding values of y. Here are the calculations:
Years Balance
5 y = 1000(1 + 0.0925/12)^(12*5)
10 y = 1000(1 + 0.0925/12)^(12*10)
15 y = 1000(1 + 0.0925/12)^(12*15)
20 y = 1000(1 + 0.0925/12)^(12*20)
25 y = 1000(1 + 0.0925/12)^(12*25)
30 y = 1000(1 + 0.0925/12)^(12*30)
Calculate each of these expressions using a calculator to find the balance for each year. Round the answers to the nearest cent.
3. To graph the equation from problem 1, you can plot the values calculated in problem 2 on a graph.
On the x-axis, plot the years from the years and balance chart (5, 10, 15, 20, 25, 30).
On the y-axis, plot the corresponding balance values for each year.
Using these points, you can connect them with a smooth curve to represent the exponential growth of the investment over time.
4. To calculate the expected value of Grace's investment if she were to invest $1000 at age 25 and not withdraw any money until age 67, you can use the exponential function from problem 1.
Substitute x = 42 (67 - 25) into the equation y = 1000(1 + 0.0925/12)^(12x) and calculate the value of y.
y = 1000(1 + 0.0925/12)^(12*42)
Calculate this expression using a calculator, and it will give you the expected value of the investment.
Consider the trend of the graph: As time increases, the balance grows exponentially, so the expected value at retirement age will likely be significantly higher than the initial investment.