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Question

A savings account earns interest by being compounded continuously. The account initially had $1,500 deposited in it. The worth of the account after t-years can be calculated using the formula: A(t)=1500e^.03t

(Part A)By what percent will the worth of the account increase per year? Round to the nearest hundredth of a percent

(Part B) To the nearest tenth of a year, how many years will it take for the worth of the account to double?

(Part C) If another investment began with a principal of $1,200 and earned interest of 4.5% applied once per year, which investment would be worth more after 10 years?

Answer 1

With Continuous Compounding, at 3% 1500 will be worth 1500*e^(.03*10) =(1.3498)*1500=2,024.78

With annual compounding, 1200 will equal 1200*(1+4.5%)¹⁰ We have to remember every year whatever was in the account earns interest and that interest which when earned will start to earn interest..

Answer 2

A) We need to figure out the impact of the continuous compounding to see how much the account will grow each year. Much like the nominal rate (APR) and actual yield (APY) on an interest bearing account at a bank, the yield will be slightly higher due to compounding and reflect the actual growth as opposed to the growth rate. Note e is a very long number and if you don't have a financial calculator with it you can punch it in with as many digits as your calculator has. e = 2.71828183

A = Pe^rt but we don't need to worry about P for this, we just need to calculate growth or e^rt. We make t = 1 for one year and we have e^r (or 2.71828183^.03) which equals 1.03045 so the account grows by 3.05% each year when rounding per the instructions.

B) 3000 = 1500e^.03t [Divide both sides by 1500]

2 = e^.03t [To isolate t we have to take the natural logarithm for each side (The natural logarithm of e is 1 and the natural logarithm of 2 0.693)]

0,693 = .03t [Divide both sides by .03]

23.1 = t

C) First calculate the value of our original investment after 10 years using the formula from part A.

A = Pe^rt

A = 1500e^.03*10

A = 1500e^.3

A = 2,024.79

Next calculate the value of the second investment using the formula for annual compounding.

A = P(1+r)^t

A = 1200(1+.045)¹⁰

A = 1,863.56

The first investment would be worth more after 10 years.

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