Problem 20 Hard Difficulty

(a) If 1000 is borrowed at $ 8\% $ interest, find the amounts due at the end of 3 years if the interest is compounded
(i) annually, (ii) quarterly, (iii) monthly, (iv) weekly,
(v) daily, (vi) hourly, and (vii) continuously.
(b) Suppose 1000 is borrowed and the interest is compounded continuously. If $ A(t) $ is the amount due after $ t $ years, where $ 0 \le t \le 3, $ graph $ A(t) $ for each of the interest rates $ 6\%, 8\%, $ and $ 10\% $ on a common screen.

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WZ

Wen Z.

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Video Transcript

here we have a compound interest problem, and we're going to use our non continuously compound interest formula for the majority of part A. The amount invested is $1000 for a time period of three years, with an interest rate of 8% and that translates into a decimal value of 0.8 And so we're going to go through several different amounts of calm poundings per year, starting with annually once per year. We use an equals one, and we get $1259.71. Then, for quarterly, we use an equals four. We substitute that into our formula. We get $1268.24 for monthly and equals 12 and that gives us $1270.24 for weekly and equals 52. That gives us $1271.1. We see the amount of money is growing, but the rate of growth is slowing as we accumulate more calm poundings for 365 compound ings per year, that would be daily. We get $1271.22 an hourly that would be 8760 compound ings per year. We get $1271.25 and for the last part, we're going to do continuous compounding, so we use a different formula. The continuous compounding formula is a equals a not e to the R. T. So we still have the same interest rate point await. We still have the same amount of time three years, and we still have the same initial investment, and we end up with $1271.25. Finally, for Part B, we want to graph the continuously compounded function for different amounts of interest rates. So 6%. Let's try that again. We have 6% we have 8% and we have 10%. So let's grab a graphing calculator. We type those in, and then we're going to look at them from time zero to time three so we can go to window and we can change our window dimensions to go from 0 to 3 on the X axis. And then I chose negative 100 to 1600 for my Y axis and then looking at the graphs. The lowest one logically is the one that has a 6% interest rate, and the middle one is the one with 8% interest rate, and the highest one is the one with a 10% interest rate.