Final answer:
To have $4,000 in 3 years with a 6% interest rate compounded quarterly, approximately $3,345.95 needs to be invested today. None of the given multiple-choice options are correct.
Step-by-step explanation:
The goal is to determine how much must be invested today to have $4,000 in 3 years with an interest rate of 6% compounded quarterly. We use the future value formula for compound interest:
FV = PV (1 + r/n)^(nt)
Here, FV is the future value ($4,000), PV is the present value (what we're solving for), r is the annual interest rate (0.06), n is the number of times the interest is compounded per year (4 for quarterly), and t is the time in years (3).
Plugging in the values, we get:
4000 = PV (1 + 0.06/4)^(4*3)
PV = 4000 / (1 + 0.06/4)^(4*3)
Now, calculate the denominator:
(1 + 0.06/4)^(4*3) = (1 + 0.015)^12 = 1.19561864852
Now, divide $4,000 by this number to find the PV:
PV = 4000 / 1.19561864852
PV = 3345.95 (approximately)
So, none of the options a. 5,718 b. 5,800 c. 5,650 d. 5,900 match the correct amount that must be invested today, which is approximately $3,345.95.