To solve this problem, we need to determine the present value of an annuity. The annuity in this case is $14,000 per year for 2 years, and the interest rate is 4.00% compounded annually.

The formula for the present value of an annuity is:

$$ PV = PMT \times \left(1 - (1 + r)^{-n}\right) / r $$

Where:
- $ PV $ is the present value of the annuity.
- $ PMT $ is the annual payment ($14,000).
- $ r $ is the annual interest rate (4.00% or 0.04).
- $ n $ is the number of years (2).

Let's plug in the values:

$$ PV = 14000 \times \left(1 - (1 + 0.04)^{-2}\right) / 0.04 $$

First, calculate $ (1 + 0.04)^{-2} $:

$$ (1 + 0.04)^{-2} = 1.04^{-2} \approx 0.924556 $$

Next, calculate $ 1 - 0.924556 $:

$$ 1 - 0.924556 = 0.075444 $$

Now, divide by the interest rate (0.04):

$$ 0.075444 / 0.04 = 1.8861 $$

Finally, multiply by the annual payment ($14,000):

$$ PV = 14000 \times 1.8861 \approx 26405.40 $$

Therefore, Austin had to deposit approximately $26,405.40 in the investment fund.

Question

To solve this problem, we need to determine the present value of an annuity. The annuity in this case is $14,000 per year for 2 years, and the interest rate is 4.00% compounded annually.

The formula for the present value of an annuity is:

$$ PV = PMT \times \left(1 - (1 + r)^{-n}\right) / r $$

Where:
- $ PV $ is the present value of the annuity.
- $ PMT $ is the annual payment ($14,000).
- $ r $ is the annual interest rate (4.00% or 0.04).
- $ n $ is the number of years (2).

Let's plug in the values:

$$ PV = 14000 \times \left(1 - (1 + 0.04)^{-2}\right) / 0.04 $$

First, calculate $ (1 + 0.04)^{-2} $:

$$ (1 + 0.04)^{-2} = 1.04^{-2} \approx 0.924556 $$

Next, calculate $ 1 - 0.924556 $:

$$ 1 - 0.924556 = 0.075444 $$

Now, divide by the interest rate (0.04):

$$ 0.075444 / 0.04 = 1.8861 $$

Finally, multiply by the annual payment ($14,000):

$$ PV = 14000 \times 1.8861 \approx 26405.40 $$

Therefore, Austin had to deposit approximately $26,405.40 in the investment fund.

Knowee AI · Accepted Answer