Questions: A company needs 6,800,000 in 9 years in order to expand their factory. How much should the company invest each week if the investment earns a rate of 8.1% compounded weekly?

Solution Steps

To solve this problem, we need to use the formula for the future value of an annuity due to weekly investments. The formula for the future value of an annuity compounded periodically is:

\[ FV = P \times \frac{(1 + r)^n - 1}{r} \]

Where:

• \( FV \) is the future value ($6,800,000)
• \( P \) is the weekly investment
• \( r \) is the weekly interest rate
• \( n \) is the total number of compounding periods (weeks)

We need to solve for \( P \):

\[ P = \frac{FV \times r}{(1 + r)^n - 1} \]

Given:

• Annual interest rate = 8.1%
• Compounded weekly, so weekly interest rate \( r = \frac{8.1\%}{52} \)
• Number of years = 9, so total weeks \( n = 9 \times 52 \)

We are given the following values:

• Future value (\(FV\)) = \$6,800,000
• Annual interest rate = 8.1%
• Number of years = 9
• Compounded weekly

The annual interest rate is 8.1%, which we convert to a decimal: \[ \text{Annual interest rate} = 0.081 \]

Since the interest is compounded weekly, we divide the annual interest rate by the number of weeks in a year (52): \[ r = \frac{0.081}{52} = 0.0015576923076923077 \]

The total number of weeks over 9 years is: \[ n = 9 \times 52 = 468 \]

We use the formula for the future value of an annuity to solve for the weekly investment (\(P\)): \[ FV = P \times \frac{(1 + r)^n - 1}{r} \]

Rearranging to solve for \(P\): \[ P = \frac{FV \times r}{(1 + r)^n - 1} \]

Substituting the given values into the formula: \[ P = \frac{6{,}800{,}000 \times 0.0015576923076923077}{(1 + 0.0015576923076923077)^{468} - 1} \]

Calculating the result: \[ P \approx 9882.440993884184 \]

Final Answer