Use x=1+r
, where r is the interest rate paid each year. Write a model polynomial, C(x)
, to represent the final amount of each of the following 7-year term investments.
Investment 1: Deposit $4,000 at the beginning of the first year.
Investment 2: Deposit $2,000 at the beginning of the first year, $1,500 at the beginning of the third year, and $500 at the beginning of the fifth year.
Using the models that you just created, which investment option will result in more interest earned at the end of the 7-year term if the annual interest rate is 3.7 percent?
Based on the answer you got for this problem, when having the same interest rate and investment term, do you think it will be better to invest more at one time or divide the investment into smaller deposits each year? Explain your reasoning.
(4 points)
For Investment 1, the model polynomial is:
C(x) = 4000(1+r)^7
For Investment 2, the model polynomial is:
C(x) = 2000(1+r)^7 + 1500(1+r)^5 + 500(1+r)^3
To compare the interest earned at the end of the 7-year term for an annual interest rate of 3.7 percent, we substitute r = 0.037 into the model polynomials.
For Investment 1:
C(x) = 4000(1+0.037)^7 = 4000(1.037)^7 ≈ 4889.56
For Investment 2:
C(x) = 2000(1+0.037)^7 + 1500(1+0.037)^5 + 500(1+0.037)^3 ≈ 5020.26
Therefore, Investment 2 will result in more interest earned at the end of the 7-year term if the annual interest rate is 3.7 percent.
When having the same interest rate and investment term, it will be better to invest more at one time rather than dividing the investment into smaller deposits each year. This is because compounding interest benefits from investing larger amounts upfront. In Investment 1, where $4,000 is deposited at the beginning of the first year, the final amount is higher than in Investment 2, where smaller amounts are deposited at different intervals.
To create the model polynomial, we need to substitute the given investment amounts into the equation x = 1 + r.
For Investment 1:
Deposit $4,000 at the beginning of the first year.
We can plug in the value of x = 4,000 into the equation: 4,000 = 1 + r.
Solving for r, we get r = 4,000 - 1 = 3,999.
Therefore, the model polynomial for Investment 1 is C(x) = x^7 = (4,000 + 3,999)^7.
For Investment 2:
Deposit $2,000 at the beginning of the first year, $1,500 at the beginning of the third year, and $500 at the beginning of the fifth year.
We need to calculate the value of x for each deposit.
For the first deposit: 2,000 = 1 + r, which gives us r = 2,000 - 1 = 1,999.
For the second deposit: 1,500 = 1 + r, which gives us r = 1,500 - 1 = 1,499.
For the third deposit: 500 = 1 + r, which gives us r = 500 - 1 = 499.
Now we can calculate the model polynomial for Investment 2 using these values of r for each deposit:
C(x) = x^7 = (2,000 + 1,999)^7 * (1,500 + 1,499)^4 * (500 + 499)^2.
To compare which investment option will result in more interest earned at the end of the 7-year term with an annual interest rate of 3.7 percent, we need to evaluate the model polynomials for each option.
For Investment 1:
C(x) = (4,000 + 3,999)^7 = 7,999^7 ≈ 1.76 * 10^30.
For Investment 2:
C(x) = (2,000 + 1,999)^7 * (1,500 + 1,499)^4 * (500 + 499)^2 = 3,999^7 * 2,999^4 * 999^2 ≈ 1.37 * 10^31.
Therefore, Investment 2 will result in more interest earned at the end of the 7-year term at a 3.7 percent annual interest rate.
When considering the same interest rate and investment term, it will be better to invest more at one time rather than dividing the investment into smaller deposits each year. As we can see from the calculations above, Investment 2 results in a significantly higher final amount than Investment 1. This is because the compounding effect of a higher initial deposit leads to a larger base for future growth and hence more interest earned overall.
To find the final amount for each investment option, we can use the model polynomial C(x) = x^n, where x represents the initial investment amount and n represents the number of years.
Investment 1:
Deposit = $4,000
The model polynomial becomes C(x) = (1 + r)^7, with x = 4000 and n = 7. Substitute these values into the equation:
C(4000) = (1 + 0.037)^7
C(4000) = 1.2745
Investment 2:
Deposit = $2,000 at the beginning of the first year, $1,500 at the beginning of the third year, and $500 at the beginning of the fifth year.
The model polynomial becomes C(x) = (1 + r)^7, with x = 2000 + 1500 + 500 and n = 7. Substitute these values into the equation:
C(4000) = (1 + 0.037)^7
C(4000) = 1.2745
Both investment options result in the same final amount, which is 1.2745 times the initial investment.
Now, let's analyze which investment option will result in more interest earned at the end of the 7-year term if the annual interest rate is 3.7 percent.
For both investment options, the interest earned is the final amount minus the initial investment. Therefore, the interest earned for both options will be the same:
Interest earned = Final amount - Initial investment
Interest earned = 1.2745 * Initial investment - Initial investment
Since the interest earned is the same for both investment options, we can conclude that it doesn't matter if you invest more at one time or divide the investment into smaller deposits each year if the interest rate and investment term are the same.
In summary, both investment options will result in the same amount of interest earned after 7 years with an annual interest rate of 3.7 percent. Whether it is better to invest more at one time or divide the investment into smaller deposits each year depends on personal preferences, financial goals, and investment strategies.