Question
Christina invests $2,000 into an account that will double every 8 years. The total amount in Christina’s investment account is modeled by the function f(x)=2,000⋅2x
, where x is time in multiples of 8 years. Which of the following is the best estimate for the total amount in the investment account if Christina invests her money for 12 years?
Option #1: $4,000
Option #2: $5,500
Option #3: $6,000
Option #4: $8,000
(1 point)
The best estimate for the investment’s total amount is Option #
, where x is time in multiples of 8 years. Which of the following is the best estimate for the total amount in the investment account if Christina invests her money for 12 years?
Option #1: $4,000
Option #2: $5,500
Option #3: $6,000
Option #4: $8,000
(1 point)
The best estimate for the investment’s total amount is Option #
Answers
GPT-4o mini
To estimate the total amount in Christina's investment account after 12 years, we will first determine how many complete 8-year periods fit into 12 years.
12 years can be divided into 8 years and the remaining 4 years:
- In 12 years, there is 1 complete 8-year period (8 years) and a remainder of 4 years.
Now, we can calculate the total value of the investment after 8 years:
- After 8 years, the investment doubles:
\[
f(1) = 2000 \cdot 2^1 = 2000 \cdot 2 = 4000
\]
Next, we need to determine how much the investment grows in the next 4 years. Since the investment doubles every 8 years, we could find its value after 4 years, but the doubling doesn't occur until the full 8 years are completed. Thus, we should consider the growth rate over the intervening years.
Since 4 years is half of the 8-year doubling period, we can consider that the investment will have increased but not fully doubled. Since the function models doubling, we can use it for fractional parts. The formula, when adjusted for fractional periods, becomes more complicated, but typically we might say the investment will grow to about 1.5 times its value after 8 years, in approximate math since it does not reach to double yet.
To estimate the future value, we can infer that since the return after half the period will be about 1.5 times the initial amount reached after the 8-year period.
So after 8 years:
- Value = $4,000
In the next 4 years, we may approximate growth. Though investors expect about 1-1.5 times. However, simply stating that division shows a better route we can explicitly apply techniques rather than half duration but typically around that, let’s look to the nearest option.
Thus, from benefit and greater accuracy, we know:
If we assume 1-1.5 gives approx
So:
- After 4 additional years, let’s consider growth near of initial stage.
Final approximate = $4000 to about around $6000 in the total for contextual reference.
Hence rounded and reasonable value folds:
The best estimate for the investment’s total amount after 12 years is:
**Option #3: $6,000**.
12 years can be divided into 8 years and the remaining 4 years:
- In 12 years, there is 1 complete 8-year period (8 years) and a remainder of 4 years.
Now, we can calculate the total value of the investment after 8 years:
- After 8 years, the investment doubles:
\[
f(1) = 2000 \cdot 2^1 = 2000 \cdot 2 = 4000
\]
Next, we need to determine how much the investment grows in the next 4 years. Since the investment doubles every 8 years, we could find its value after 4 years, but the doubling doesn't occur until the full 8 years are completed. Thus, we should consider the growth rate over the intervening years.
Since 4 years is half of the 8-year doubling period, we can consider that the investment will have increased but not fully doubled. Since the function models doubling, we can use it for fractional parts. The formula, when adjusted for fractional periods, becomes more complicated, but typically we might say the investment will grow to about 1.5 times its value after 8 years, in approximate math since it does not reach to double yet.
To estimate the future value, we can infer that since the return after half the period will be about 1.5 times the initial amount reached after the 8-year period.
So after 8 years:
- Value = $4,000
In the next 4 years, we may approximate growth. Though investors expect about 1-1.5 times. However, simply stating that division shows a better route we can explicitly apply techniques rather than half duration but typically around that, let’s look to the nearest option.
Thus, from benefit and greater accuracy, we know:
If we assume 1-1.5 gives approx
So:
- After 4 additional years, let’s consider growth near of initial stage.
Final approximate = $4000 to about around $6000 in the total for contextual reference.
Hence rounded and reasonable value folds:
The best estimate for the investment’s total amount after 12 years is:
**Option #3: $6,000**.