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.
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.
To represent the final amount of a 7-year investment using the given information, we will need to calculate the compound interest for each year and sum them up.
First, let's define the polynomial c(x) using the given formula x = 1 + r, where r is the interest rate paid each year:
c(x) = (1 + r)
Now, let's break down the investment into three parts:
1. The initial deposit of $5,000 at the beginning of the first year:
This amount will accumulate interest for seven years, so we can represent it as:
c1(x) = c(x)^7 * 5000
2. The deposit of $2,000 at the beginning of the third year:
This amount will accumulate interest for five years (from the third to the seventh year), so we can represent it as:
c2(x) = c(x)^5 * 2000
3. The remaining four years without any deposits:
For these four years, we only need to consider the interest on the previous investments, so no additional terms need to be added.
Finally, we can represent the total amount of the investment (final amount) as the sum of the three parts:
c(x) = c1(x) + c2(x)
Substituting the expressions for c1(x) and c2(x) into the equation, we get:
c(x) = (1 + r)^7 * 5000 + (1 + r)^5 * 2000
This is the model polynomial c(x) that represents the final amount of a 7-year investment with the given deposit amounts at specific time intervals.
To build the polynomial model, we need to break down the problem into different time periods and calculate the value of the investment at the end of each period.
Let's start by identifying the relevant time periods:
1. The first year: $5,000 is deposited at the beginning.
2. The second year: No additional deposits.
3. The third year: $2000 is deposited at the beginning.
4. Fourth year and beyond: No additional deposits.
Now, let's break down the calculation for each time period:
1. First year:
- Initial deposit: $5,000
- Interest earned: $5,000 * r (since the interest rate is applied to the initial investment)
- Total amount at the end of the first year: $5,000 + $5,000 * r
2. Second year:
- Initial amount at the beginning of the second year: $5,000 + $5,000 * r
- Interest earned: ($5,000 + $5,000 * r) * r (since the interest rate is applied to the previous year's total)
- Total amount at the end of the second year: ($5,000 + $5,000 * r) + ($5,000 + $5,000 * r) * r
3. Third year:
- Initial deposit: $2,000
- Interest earned: $2,000 * r
- Total amount at the end of the third year: ($5,000 + $5,000 * r) + ($5,000 + $5,000 * r) * r + $2,000 + $2,000 * r
4. Fourth year and beyond:
- No additional deposits made during this period.
- The interest earned each year is applied to the previous year's total.
Based on the above calculations, we can construct the polynomial model c(x) representing the final amount of the 7-year investment:
c(x) = ($5,000 + $5,000 * r) + ($5,000 + $5,000 * r) * r + $2,000 + $2,000 * r
Simplifying the polynomial, we get:
c(x) = $12,000 + $12,000 * r + $10,000 * r^2
Therefore, the final amount of the investment after 7 years, taking into account the given deposits, is represented by the polynomial c(x) = $12,000 + $12,000 * r + $10,000 * r^2.