To determine how much a $240 investment is worth after 14 years with an interest rate of 9% per year compounded monthly, we need to use the compound interest formula. The compound interest formula is given by:
[tex]\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \][/tex]
Where:
- \( A \) is the amount of money accumulated after n years, including interest.
- \( P \) is the principal amount (the initial amount of money).
- \( r \) is the annual interest rate (decimal).
- \( n \) is the number of times that interest is compounded per year.
- \( t \) is the time the money is invested for, in years.
Given:
- \( P = 240 \)
- \( r = 9\% = 0.09 \)
- \( n = 12 \) (since the interest is compounded monthly)
- \( t = 14 \) years
Now, plug these values into the compound interest formula:
[tex]\[ A = 240 \left(1 + \frac{0.09}{12}\right)^{12 \times 14} \][/tex]
First, calculate \(\frac{0.09}{12}\):
[tex]\[ \frac{0.09}{12} = 0.0075 \][/tex]
Next, add 1 to the result:
[tex]\[ 1 + 0.0075 = 1.0075 \][/tex]
Then, multiply the number of times interest is compounded per year by the number of years:
[tex]\[ 12 \times 14 = 168 \][/tex]
Raise the base to the power of this product:
[tex]\[ 1.0075^{168} \approx 3.508885595 \][/tex]
Finally, multiply this result by the principal amount to find the future value:
[tex]\[ A = 240 \times 3.508885595 \approx 842.13 \][/tex]
Therefore, the investment will be worth approximately \$842.13 in 14 years.
For the given multiple-choice options, the correct answer is:
[tex]\[ \boxed{\$842.13} \][/tex]
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Now, let's address the value of the sailboat after 7 years, given annual depreciation of 15%.
The depreciation formula used is:
[tex]\[ V = P(1 - r)^t \][/tex]
Where:
- \( V \) is the value of the sailboat after depreciation
- \( P \) is the initial value of the sailboat
- \( r \) is the annual depreciation rate (as a decimal)
- \( t \) is the time in years
Given:
- \( P = 9500 \)
- \( r = 15\% = 0.15 \)
- \( t = 7 \) years
Substitute these values into the depreciation formula:
[tex]\[ V = 9500 \times (1 - 0.15)^7 \][/tex]
First, calculate \( 1 - 0.15 \):
[tex]\[ 1 - 0.15 = 0.85 \][/tex]
Next, raise this to the 7th power:
[tex]\[ 0.85^7 \][/tex]
Calculate \( 0.85^7 \):
[tex]\[ 0.85^7 \approx 0.3119501125 \][/tex]
Finally, multiply this result by the initial value of the sailboat:
[tex]\[ V = 9500 \times 0.3119501125 \approx 2963.53 \][/tex]
Therefore, the value of the sailboat after 7 years is approximately \$2963.53.
