Question

Isabel needs $29,260 for a future project. she can invest $5000 now at an annual rate of 11%, compounded semiannually. assuming that no withdrawals are made, how long will it take for her to have enough money for her project?

Answer 1

Isabel will need approximately 8.14 years to have enough money for her project.

Step-by-step explanation:

To determine how long it will take for Isabel to have enough money for her project, we can use the compound interest formula:

A = P(1 + r/n)^(nt)

Where:

• A is the future amount
• P is the present amount (initial investment)
• r is the annual interest rate in decimal form
• n is the number of times interest is compounded per year
• t is the number of years

In this case, Isabel needs $29,260, so A = $29,260. Her initial investment is $5000, so P = $5000. The annual interest rate is 11% or 0.11, compounded semiannually, so r = 0.11 and n = 2. We need to find t.

Substituting the values into the formula:

$29,260 = $5000(1 + 0.11/2)^(2t)

Divide both sides by $5000:

5.852 = (1 + 0.055)^2t

Take the natural logarithm of both sides to eliminate the exponent:

ln(5.852) = ln((1 + 0.055)^2t)

We can solve for t by dividing both sides by ln((1 + 0.055)^2) and multiplying by the reciprocal of 2:

t = ln(5.852) / (2 * ln(1.055))

Using a calculator, we find that t ≈ 8.14 years.