The function f(t)=12,000(1.075)t models the value of an investment t years from now. What is the meaning of the value of f(5)?
The Correct Answer and Explanation is :
The correct answer is: 17,284.80
The function ( f(t) = 12,000(1.075)^t ) models the future value of an investment of $12,000 compounded annually at an interest rate of 7.5%. Here, ( t ) represents the number of years the money is invested, and ( f(t) ) represents the future value of the investment after ( t ) years.
To find the value of ( f(5) ), we substitute ( t = 5 ):
[
f(5) = 12,000(1.075)^5
]
Calculating ( (1.075)^5 ):
[
(1.075)^5 \approx 1.4404
]
Now, we can find ( f(5) ):
[
f(5) = 12,000 \times 1.4404 \approx 17,284.80
]
Thus, ( f(5) \approx 17,284.80 ).
Explanation
The value of ( f(5) ), approximately $17,284.80, represents the total worth of the initial $12,000 investment after five years, given the annual compounding interest of 7.5%. This amount includes the initial principal plus the interest accrued over the five years.
To understand this in context, it is essential to consider the effect of compound interest. Unlike simple interest, which is calculated only on the principal amount, compound interest calculates interest on the principal and any interest that has previously been added to the investment. This means that over time, the investment grows at an increasing rate, as each year’s interest is calculated on a progressively larger amount.
In practical terms, if an investor were to withdraw this amount after five years, they would have significantly more than their original investment. This illustrates the power of compound interest in growing investments over time, highlighting the importance of early and sustained investment for achieving financial goals. In summary, ( f(5) ) provides insight into how a consistent investment strategy can yield substantial returns over time.