Question

3. You want to have $4000 in your savings account after 2 years. Find the amount you should deposit for each of the situations described below. a. The account pays 3% annual interest compounded monthly. b. The account pays 4% annual interest compounded continuously.​

Answer 1

Part A)

About $3767.34.

Part B)

About $3692.47.

Explanation:

Part A)

Recall that compound interest is given by the formula:

`\displaystyle A = P\left(1+(r)/(n)\right)^(nt)`

Where A is the final amount, P is the initial amount, r is the interest rate, n is the number of times compounded per year, and t is the number of years.

To obtain $4000 after two years, let A = 4000 and t = 2.

Because the account pays 3% interest compounded monthly, r = 0.03 and n = 12.

Substitute and solve for P:

`\displaystyle \begin{aligned} (4000) & = P\left(1+((0.03))/((12))\right)^((12)(2)) \\ \\ P & = (4000)/(\left(1+((0.03))/((12))\right)^((12)(2))) \\ \\ & \approx \$3767.34\end{aligned}`

In concluion, about $3767.34 should be deposited.

Part B)

Recall the formula for continuous compound:

`\displaystyle A = Pe^(rt)`

Where e is Euler's number.

Hence, let A = 4000, r = 0.04 and t = 2. Substitute and solve for P:

`\displaystyle \begin{aligned}(4000) & = Pe^((0.04)(2)) \\ \\ P & = (4000)/(e^((0.02)(4))) \\ \\ & \approx \$3692.47 \end{aligned}`

In conclusion, about $3692.47 should be deposited.