WEBVTT
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here we have a compound interest problem and we're going
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to do part a mostly with the non continuous compounding
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formula. And so we have, in an amount
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invested$3000 an interest rate of 5%. So we
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use the decimal point verify and a time of five
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years. So to find the final amount, if
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interest is calm, pounded annually, which would be
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an equals 11 time per year, we have a
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equals 3000 times one plus 0.0 5/1 raised to the
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power one times five. And we put that in
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a calculator and we get rounded to the nearest penny
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.$3828.84. Now we move on to the second
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part of part A, and we're going to calculate
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it for semi annually, which would be two times
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a year so and equals two. So a equals
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3000 times one plus 0.0 5/2 raised to the two
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times five, which would be 10 10 compound ings
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in the five year period. And we put that
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in a calculator and rounding to the nearest penny we
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get$3840.25 for the third part of part. They
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were going to calculate for monthly compounding, so that
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would be 12 times of compounding per year. So
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we have a equals 3000 times one plus 0.0 5/12
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raised to the power 12 times five, which would
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be 60 compound ing's and rounded to the nearest penny
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. We get$3850 and 20 and ah, seven
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cents seven cents. So we could see the amounts
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are going up a little bit each time as the
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number of compound ings increases for the fourth part were
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compounding weekly. There are 52 weeks in a year
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, so an equals 52. So we have a
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gal's 3000 times one plus 0.0 5/52 raised to the
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power 52 times five, and that's going to give
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us$3851.61. And for the fifth part of part
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, a daily compounding will use n equals 365.
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So we have a equals 3000 times one plus 0.0
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5/3 65 raised to the power 3 65 times five
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. Put that into a calculator and we get$3852.1
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. And for part six, we're going to switch
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to continuously compounded interest. So we have a different
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model for this. We have a equals a not
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times e to the R. T. So that's
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going to be 3000 times E to the 0.5 times
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five, and we end up with$3852.8 not a
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lot more than the previous one. But it's the
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highest you can get for continuously compounded interest compared to
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any of the other calm poundings now for Part B
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were to find the differential equation that fits the situation
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. So because we have this basic exponential growth type
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model here, it fits. The differential equation rate
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of change of A is proportional to a so rate
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of change of a equals K times A. Now
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we know the value of K will be our rate
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. And in our equation we had a equals 3000
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times E to the 0.5 t, so the rate
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was your a 0.0 fight. So that's R K
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. So now we have de a d t equals
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0.5 times a, and we can go ahead and
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substitute our equation. And for a and we have
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0.5 times 3000 each of the 0.5 t. All
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right, let's multiply the 0.5 and the 3000 together
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, and we get 1 50 e to the 0.5
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t. So that's our differential equation for this situation
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. And for the initial condition. That would be
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what's the amount when the time is zero and we
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know the amount invested was$3000. So at time
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zero the amount is 3000. That's the initial condition
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.