Question

A person invests 4500 dollars in a bank. The bank pays 5.75% interest compounded semi-annually. To the nearest tenth of a year, how long must the person leave the money in the bank until it reaches 10000 dollars?

Answer 1

Around 10.6 years

Explanation:

5.75% of 4500 is 258.75

then we will need to subtract the current amount from the final amount.

10000-4500=5500

next

5500/258.75=21.256

finally

21.256/2=10.628

Answer 2

With a $4,500 investment at 5.75% interest compounded semi-annually, it takes approximately 7.9 years for the money to grow to $10,000.

To determine how long it takes for the investment to reach $10,000 with a 5.75% interest rate compounded semi-annually, we can use the compound interest formula:

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

Where:

-
`\(A\)` is the future value of the investment
`(\$10,000)`,

-
`\(P\)` is the principal amount
`(\$4,500)`,

-
`\(r\)` is the annual interest rate (5.75% or 0.0575 as a decimal),

-
`\(n\)` is the number of times interest is compounded per year (2 for semi-annual),

-
`\(t\)` is the time the money is invested for in years.

Rearranging the formula to solve for
`\(t\)`:

`\[t = (\log(A/P))/(n \cdot \log(1 + (r)/(n)))\]`

Now, we can substitute the given values:

`\[t = (\log(10,000/4,500))/(2 \cdot \log(1 + (0.0575)/(2)))\]`

Calculating this expression gives approximately
`\(t \approx 7.9\)` years.

Therefore, to the nearest tenth of a year, the person must leave the money in the bank for approximately 7.9 years for it to reach $10,000.