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College Algebra Unit 7 Logarithm and Exponents Review Name: 13. Doubling Time for Continuous Compounding: An investment is made in a trust fund at an annu interest rate of 7.75%, compounded continuously. How long will it take for the investment to double?

User Sherece
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In order to solve this problem, we will use the formula for continuous compounding, which is A = Pe^(rt), where:

- A is the final amount,
- P is the principal amount (the initial amount of money),
- r is the annual interest rate, and
- t is the time in years.

We are given an initial investment P, an interest rate r of 7.75%, and we want to find out the time it takes for the investment to double. So, we want A to be twice the initial amount, that is, A = 2P.

Substituting A = 2P into the initial formula gives us the equation 2P = Pe^(rt). We can simplify this equation by dividing both sides by P, which gives us:

2 = e^(rt)

Next, we need to solve this equation for t. To do this, we'll use a logarithm to eliminate the exponential function e^x.

By applying the natural logarithm ln to both sides of the equation, we obtain:

ln(2) = rt.

And finally, rearranging to solve for t gives us:

t = ln(2) / r.

Before substituting our values, we have to convert our annual interest rate from percentage to a decimal. The rate r is thus 7.75 / 100 = 0.0775.

Substituting r into our equation yields:

t = ln(2) / 0.0775.

Evaluating this expression, we find

t ≈ 8.944.

This is the number of years it will take for the investment to double under continuous compounding at an annual interest rate of 7.75%.

User Muhammad Ali
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