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Jwantsto save up $12000 fora vacationHe decided to invest $3000 into high saving account that offer an annual compound interest rate of 4%How many years takeforJ toreach goal if compound semiannually?

Jan 13

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Jan 13

To find out how many years it will take for J to reach his goal of $12,000, we can use the compound interest formula:

A = P(1 + r/n)^(nt)

Where:

A = the future value of the investment

P = the principal amount (initial investment)

r = annual interest rate (as a decimal)

n = number of times interest is compounded per year

t = number of years

In this case, J invested $3,000 at an annual compound interest rate of 4% (or 0.04), and the interest is compounded semiannually (n = 2).

$12,000 = $3,000(1 + 0.04/2)^(2t)

Dividing both sides of the equation by $3,000:

4 = (1 + 0.02)^(2t)

Taking the natural logarithm of both sides:

ln(4) = ln((1 + 0.02)^(2t))

Using the property of logarithms, we can bring down the exponent:

ln(4) = 2t * ln(1 + 0.02)

Dividing both sides by 2 * ln(1 + 0.02):

t = ln(4) / (2 * ln(1 + 0.02))

Using a calculator, we can find:

t ≈ 17.67

Therefore, it will take approximately 17.67 years for J to reach his goal if the interest is compounded semiannually.

A = P(1 + r/n)^(nt)

Where:

A = the future value of the investment

P = the principal amount (initial investment)

r = annual interest rate (as a decimal)

n = number of times interest is compounded per year

t = number of years

In this case, J invested $3,000 at an annual compound interest rate of 4% (or 0.04), and the interest is compounded semiannually (n = 2).

$12,000 = $3,000(1 + 0.04/2)^(2t)

Dividing both sides of the equation by $3,000:

4 = (1 + 0.02)^(2t)

Taking the natural logarithm of both sides:

ln(4) = ln((1 + 0.02)^(2t))

Using the property of logarithms, we can bring down the exponent:

ln(4) = 2t * ln(1 + 0.02)

Dividing both sides by 2 * ln(1 + 0.02):

t = ln(4) / (2 * ln(1 + 0.02))

Using a calculator, we can find:

t ≈ 17.67

Therefore, it will take approximately 17.67 years for J to reach his goal if the interest is compounded semiannually.