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Savannah wants to start a business, which she estimates will cost $9,885.00. If Savannah puts $9,085.00 into a savings account which earns 12% interest compounded continuously, how long will it be until she has enough money to start the business?

Question

Savannah wants to start a business, which she estimates will cost 9,885.00.IfSavannahputs9,885.00. If Savannah puts 9,085.00 into a savings account which earns 12% interest compounded continuously, how long will it be until she has enough money to start the business?

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Solution

To solve this problem, we can use the formula for continuous compound interest, which is A = Pe^(rt), 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 (in decimal).
  • t is the time the money is invested for, in years.

We know that:

  • A = $9,885.00 (the amount Savannah needs to start her business).
  • P = $9,085.00 (the initial amount Savannah has).
  • r = 12% = 0.12 (the annual interest rate).
  • t is what we're trying to find.

We can plug the known values into the formula and solve for t:

9,885.00=9,885.00 = 9,085.00 * e^(0.12t)

To isolate the exponential term, we divide both sides by $9,085.00:

9,885.00/9,885.00 / 9,085.00 = e^(0.12t)

Then, to get rid of the base e, we can take the natural logarithm (ln) of both sides:

ln(9,885.00/9,885.00 / 9,085.00) = ln(e^(0.12t))

The natural logarithm and the base e cancel each other out on the right side of the equation, leaving:

ln(9,885.00/9,885.00 / 9,085.00) = 0.12t

Finally, to solve for t, we divide both sides by 0.12:

t = ln(9,885.00/9,885.00 / 9,085.00) / 0.12

Now, you just need to calculate the value of the natural logarithm and divide by 0.12 to find the time, in years, it will take for Savannah to have enough money to start her business.

This problem has been solved

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