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.
1. To write an exponential function that models this situation, we need to use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A is the final amount including interest
P is the principal amount (initial investment)
r is the annual interest rate (as a decimal)
n is the number of times that interest is compounded per year
t is the time in years
Given that Grace deposits $1000 in a mutual fund earning 9.25% annual interest, compounded monthly, we have:
P = $1000
r = 9.25% = 0.0925 (as a decimal)
n = 12 (compounded monthly)
Replacing these values in the formula, we get the exponential function:
y(x) = 1000(1 + 0.0925/12)^(12x)
2. To complete the years and balance chart, we can evaluate the function for different values of x (years) and round the results to the nearest cent:
Years Balance
5 $1,563.98
10 $2,598.85
15 $4,316.95
20 $7,165.22
25 $11,896.10
30 $19,787.60
3. To graph the equation y(x) = 1000(1 + 0.0925/12)^(12x), we can plot the values from the balance chart on a graph with an x-scale of 5 years and a y-scale of $1000. Each point represents a (year, balance) pair.
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 can substitute x = 42 (67 - 25) into the exponential function:
y(42) = 1000(1 + 0.0925/12)^(12 * 42)
y(42) = $305,461.65
So, the expected value of her investment is $305,461.65. The graph shows that the balance increases significantly over time, indicating the long-term potential for growth.