Questions: You can afford monthly deposits of 300 into an account that pays 4.2% compounded monthly. How long will it be until you have 8,900 to buy a boat? Type the number of months: (Round to the next-higher month if not exact.)

You can afford monthly deposits of 300 into an account that pays 4.2% compounded monthly. How long will it be until you have 8,900 to buy a boat?

Type the number of months:
(Round to the next-higher month if not exact.)

Transcript text: You can afford monthly deposits of $\$ 300$ into an account that pays $4.2 \%$ compounded monthly. How long will it be until you have $\$ 8,900$ to buy a boat? Type the number of months: $\square$ (Round to the next-higher month if not exact.)

Solution

Solution Steps

Step 1: Identify the Given Parameters

The target amount (A) is $8900, the monthly deposit (D) is $300, and the annual interest rate (r) is 4.2%. The goal is to find the number of months (n) required to reach or exceed the target amount.

Step 2: Apply the Future Value of an Annuity Formula

The future value $FV$ of a series of monthly deposits is calculated using the formula: $$FV = D \times \left( \frac{(1 + \frac{r}{12})^n - 1}{\frac{r}{12}} \right)$$ Given the target amount $A$, we rearrange the formula to solve for $n$, the number of months required.

Step 3: Rearrange the Formula to Solve for $n$

$$n = \frac{\log\left(1 + \frac{A \times \frac{r}{12}}{D}\right)}{\log\left(1 + \frac{r}{12}\right)}$$ Since the number of months must be an integer, we apply the ceiling function to $n$.

Step 4: Calculate the Number of Months

After applying the formula, the number of months required to reach or exceed the target amount is calculated to be 29.