Final answer:
Lucía should invest around $5488.46 now at 7.65% compounded quarterly to meet her goal.
Step-by-step explanation:
To find out how much money Lucia should invest now to reach her goal, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where A is the final amount, P is the principal (initial amount), r is the annual interest rate (7.65% or 0.0765), n is the number of times interest is compounded per year (4 times for quarterly compounding), and t is the number of years (6 years).
Substituting the given values into the formula:
A = P(1 + 0.0765/4)^(4*6)
7000 = P(1 + 0.019125)^(24)
Simplifying the equation:
7000 = P(1.019125)^(24)
Dividing both sides by (1.019125)^(24):
P = 7000 / (1.019125)^(24)
Using a calculator, P is approximately $5488.46. Therefore, Lucía should invest around $5488.46 now at 7.65% compounded quarterly to meet her goal.