The function f(t) = 12000 * (1.075) ^ t models the value of an investment t years from now. What is the meaning of the value of /(5)?
OA. the interest rate the investment earns
B. the value of the investment 5 years ago
C. the initial value of the investment
D. the value of the investment in 5 years
The correct answer and explanation is:
The correct answer is D. the value of the investment in 5 years.
The function f(t)=12000×(1.075)tf(t) = 12000 \times (1.075)^t models the value of an investment at time tt years from the present. Here’s the breakdown of each part of the formula:
- 12000: This represents the initial value of the investment at time t=0t = 0. It is the principal amount that was originally invested.
- (1.075)^t: This is the factor by which the investment grows each year. The base of 1.0751.075 indicates that the investment grows by 7.5% per year (since 1+0.075=1.0751 + 0.075 = 1.075).
- t: This is the number of years into the future for which the value is being calculated. As tt increases, the investment grows.
When we want to know f(5)f(5), it means we are calculating the value of the investment after 5 years. To find this, substitute t=5t = 5 into the equation: f(5)=12000×(1.075)5f(5) = 12000 \times (1.075)^5
Using a calculator, we find: f(5)≈12000×1.441f(5) \approx 12000 \times 1.441 f(5)≈17292f(5) \approx 17292
This shows that the value of the investment after 5 years is approximately $17,292.
Thus, f(5)f(5) gives the value of the investment in 5 years based on the growth formula. Therefore, option D is the correct interpretation of the function value at t=5t = 5.
Let’s rule out the other options:
- A (the interest rate) is incorrect because the interest rate is part of the formula but not the direct result of evaluating f(5)f(5).
- B (the value of the investment 5 years ago) is incorrect because f(t)f(t) gives future values, not past values.
- C (the initial value of the investment) is incorrect because the initial value, 1200012000, is not calculated by the function f(t)f(t); it is simply a part of the formula.
Thus, the correct interpretation of f(5)f(5) is the value of the investment in 5 years.