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Use x=1+r%0D%0A𝑥%0D%0A=%0D%0A1%0D%0A+%0D%0A𝑟%0D%0A, where r is the interest rate paid each year. Write a model polynomial, C(x)%0D%0A𝐶%0D%0A(%0D%0A𝑥%0D%0A)%0D%0A, to represent the final amount of each of the following 7-year term investments.%0D%0A%0D%0AInvestment 1: Deposit $4,000 at the beginning of the first year.%0D%0A%0D%0AInvestment 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.%0D%0A%0D%0AUsing 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?%0D%0A%0D%0ABased 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.

1 answer

To create the polynomial models for each of the investments, we'll start with the formula you provided:

\[ C(x) = C_0 \cdot x^n \]

Where \(C_0\) is the initial deposit amount and \(n\) is the number of years until the end of the investment period.

Investment 1:

For the first investment, $4,000 is deposited at the beginning of the first year and earns interest over the full 7 years.

  • The interest rate \(r\) is 3.7%, which in decimal form is \(0.037\).
  • Therefore, \(x = 1 + r = 1 + 0.037 = 1.037\).

The value at the end of 7 years can be modeled with: \[ C(x) = 4000 \cdot x^7 \] Substituting for \(x\): \[ C(x) = 4000 \cdot (1.037)^7 \]

Investment 2:

For the second investment, we have three deposits:

  1. $2,000 at the beginning of the first year (grows for 7 years).
  2. $1,500 at the beginning of the third year (grows for 5 years).
  3. $500 at the beginning of the fifth year (grows for 3 years).

For these, we can model their contributions separately and then sum them to get the final amount.

  1. First deposit: \[ C_1(x) = 2000 \cdot x^7 \]
  2. Second deposit: \[ C_2(x) = 1500 \cdot x^5 \]
  3. Third deposit: \[ C_3(x) = 500 \cdot x^3 \]

Thus, the total polynomial model for the second investment is: \[ C(x) = 2000 \cdot x^7 + 1500 \cdot x^5 + 500 \cdot x^3 \] Substituting for \(x\): \[ C(x) = 2000 \cdot (1.037)^7 + 1500 \cdot (1.037)^5 + 500 \cdot (1.037)^3 \]

Calculating the Final Amounts

Now we can calculate the amounts for both investments using \(x = 1.037\).

  1. For Investment 1: \[ C_1 = 4000 \cdot (1.037)^7 \]

    Using a calculator: \[ (1.037)^7 \approx 1.28368 \] Then: \[ C_1 \approx 4000 \cdot 1.28368 \approx 5134.72 \]

  2. For Investment 2:

    • For the first deposit: \[ C_1 = 2000 \cdot (1.037)^7 \approx 2000 \cdot 1.28368 \approx 2567.36 \]
    • For the second deposit: \[ C_2 = 1500 \cdot (1.037)^5 \approx 1500 \cdot 1.19974 \approx 1799.61 \]
    • For the third deposit: \[ C_3 = 500 \cdot (1.037)^3 \approx 500 \cdot 1.11364 \approx 556.82 \]

    Therefore, the total for Investment 2 is: \[ C_2 \approx 2567.36 + 1799.61 + 556.82 \approx 4923.79 \]

Comparison of Interest Earned

  • Investment 1: Approximately $5134.72
  • Investment 2: Approximately $4923.79

Interest Earned:

  • Investment 1 earns about $1134.72 in interest.
  • Investment 2 earns about $492.79 in interest.

Conclusion

Investment 1 results in more interest earned.

Reasoning About Investment Strategy

It is generally better to invest a larger sum at one time rather than spreading it out with smaller deposits, provided the interest rate remains constant. This is due to the principle of compounding interest: the sooner the money is invested, the longer it has to grow with interest. In the case of Investment 1, the entire amount is subject to compounding for the full term of the investment, whereas in Investment 2, later deposits have less time to accumulate interest.