use x=1+r, when r is the interest rate paid each year. write a model polynomial, c(x). represent the final amount of a 7 year investment if $5,000 was deposited at the beginning of the first year, and $2000 was deposited at the beginning of the third year.

Answer 1

To represent the final amount of the investment, we need to take into account the interest rate and the deposits made at different times.

Let's break down the investment into three time periods:
1. From year 1 to year 7: A deposit of $5,000 was made at the beginning of the first year, and it compounds for 7 years. Thus, the contribution to the final amount from this period is $5,000 * x^7.

2. From year 3 to year 7: A deposit of $2,000 was made at the beginning of the third year, and it compounds for 5 years. Thus, the contribution to the final amount from this period is $2,000 * x^5.

3. From year 8 onwards: No additional deposits were made after year 7, so we will only consider the compounding of the existing amount. The contribution to the final amount from this period is x raised to the power of the number of years beyond year 7, which is (x^7)^2.

Now, let's combine these three contributions together to form the polynomial model:

c(x) = $5,000 * x^7 + $2,000 * x^5 + (x^7)^2

This model polynomial represents the total amount of the investment after 7 years, taking into account the initial deposits and the interest rate.