If $3000 is deposited at the end of each half year in an account that earns 6.2% compounded semiannually, how long will it be before the account contains

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87 To find out how long it will be before the account contains a certain amount, we need to use the formula for compound interest:

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

where:
A = the future value of the account
P = the principal amount (the initial deposit)
r = the annual interest rate (in decimal form)
n = the number of times the interest is compounded per year
t = the number of years

In this case, the principal (P) is $3000, the annual interest rate (r) is 6.2% (or 0.062 as a decimal), and the interest is compounded semiannually (n = 2).

Let's assume we want the account to contain $50000. We can substitute these values into the formula and solve for t:

50000 = 3000(1 + 0.062/2)^(2t)

Now, we can solve this equation for t. First, divide both sides of the equation by 3000:

(1 + 0.062/2)^(2t) = 50000/3000

Simplify the right side:

(1 + 0.031)^(2t) = 16.6667

Next, take the logarithm of both sides with base (1 + 0.031):

log[(1 + 0.031)^(2t)] = log(16.6667)

Using log properties, we can bring the exponent down:

2t * log(1 + 0.031) = log(16.6667)

Now, divide both sides of the equation by log(1 + 0.031):

2t = log(16.6667) / log(1 + 0.031)

Finally, divide both sides by 2 and solve for t:

t = (log(16.6667) / log(1 + 0.031)) / 2

Using a scientific calculator or software, calculate the right side of the equation to find t. The result will give you the number of years it will take for the account to contain $50000.