Carl plans to invest $500 at 8.25% interest, compounded continuously. How long will it take for his money to triple?

Answer 1

To find the time it takes for the money to triple, we use the continuous compounding formula:

[ A = P \times e^{rt} ]

Where:

  • ( A ) is the amount of money after time ( t )
  • ( P ) is the principal amount (initial investment)
  • ( r ) is the annual interest rate (in decimal form)
  • ( t ) is the time in years
  • ( e ) is Euler's number (approximately 2.71828)

Given:

  • ( P = $500 )
  • ( r = 8.25% ) (convert to decimal: ( 0.0825 ))
  • We want ( A ) to be triple the initial investment, so ( A = 3P = 3 \times 500=500 = 1500 )

Substituting the values into the formula, we have:

[ 1500 = 500 \times e^{0.0825t} ]

Now, solve for ( t ):

[ \frac{1500}{500} = e^{0.0825t} ]

[ 3 = e^{0.0825t} ]

Taking the natural logarithm (ln) of both sides to isolate ( t ):

[ \ln(3) = 0.0825t ]

[ t = \frac{\ln(3)}{0.0825} ]

[ t \approx \frac{1.0986}{0.0825} ]

[ t \approx 13.31 \text{ years} ]

So, it will take approximately 13.31 years for Carl's money to triple at an interest rate of 8.25%, compounded continuously.

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Answer 2

#approx 13.32# years

A principal amount of #$P#, compounded continuously at a rate of #r%#p.a. will grow to #$Q# after #t# years is given by:
#Q=Pe^(rt)#

In our example:

P=500, r=8.25%#, Q=3xx500 = 1500#
#:. 1500=500xxe^(8.25/100xxt)#
#e^(0.085t)=3#
#0.0825t = ln3#
#t = ln3/0.0825 approx 1.0986/0.0825#
#t approx 13.32# years
{NB: The principal is actually irrelevant here. Any amount will triple in #approx 13.32# years if compounded continuously at #8.25%# p.a.]
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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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