Handout No. 5. A(t) = P e rt

Transcription

1 Name: MATH 1113 Precalculus Eric Perkerson Date: October 12, 2014 Handout No. 5 Problem 8 v.1 If P = 500 dollars is deposited in a savings account that pays interest at a rate of 4 = 19/2% per year compounded continuously, find the balance after t = 4 years. We know that for continuously compounded interest, we have At) = P e rt we plug in A4) = 500)e 19/200) Problem 8 v.2 How much money, invested at an interest rate of r = 6.6% per year compounded continuously, will amount to A = dollars after t = 13 years? Round your answer to the nearest cent.) Set and solve for = P e 6.6/100)13 P = e 6.6/100) Problem 8 v.3 Alice invests her money at Bob s bank. Bob compounds interest quarterly at a nominal rate of 6%. Alice wants the investment to be worth $7000 after 8 years. How much money should Alice invest? Set and solve for 7000 = P 1 + 6/100 ) 48) 4 P = /100 ) 48) Problem 9 v.1 Determine the per annum interest rate required for an investment with continuous compound interest to triple in 7.5 years. Express your answer as a percent. We know that an investment with continuous compound interest of r yields an amount of money at time t of At) = P e rt so the amount of money at t = 7.5 is A7.5) = P e r7.5) = P e r/2)

2 and we know that our starting amount of money is the principle, P. triple the principle is 3P. So we set 3P = P e r/2) and solve for r. Hence 3P = P e r/2) 3 = e r/2) ln3) = r/2) 2 ln3) = r r = 2 ln3) r %) = % ) 2 ln3) Problem 9 v.2 You invest $8,800 at 4% per annum compounded continuously. Determine the time T in years) for your investment to be worth $14,350. We know that an investment of $8, 800 with continuous compound interest of 4% yields an amount of money at time t of At) = )e.04)t = )e 4/100)t so if we want this to equal $14, 350 at t = T, we set and then solve for T, thus we have = )e 4/100)T = )e 4/100)T = e4/100)t ) ln = 1/25)T ) ln = T T = 25 ln )

3 Problem 10 v.1 Determine the effective interest rate if a nominal rate of 5.5% is compounded monthly. Express your answer as a percent. We know that a nominal rate of 5.5% compounded monthly yields an amount of money at time t of At) = P ) 12t = P ) 12t 12 and we know that the effective interest rate, call it r, yields an amount of money at time t of At) = P 1 + r) t Because these are both the same function, these expressions for the function must be the same for all t, so in particular they are the same for t = 1. So we set and then we solve for r, the effective rate of interest: P P ) 121) = P 1 + r) 1) ) 121) = P 1 + r) 1) P ) 12 = P 1 + r) ) 12 = 1 + r ) 12 1 = r r = ) r %) = % ) 12 1) Problem 9 v.3 Determine the per annum interest rate required for an investment with continuous compound interest to double in 7.5 years. Express your answer as a percent. We know that a per annum interest rate of r compounded continuously yields an amount of money at time t of At) = P e rt

4 so the amount of money at t = 7.5 is A7.5) = P e r7.5) = P e r/2) and we know that our starting amount of money is the principle, P. double the principle is 2P. So we set 2P = P e r/2) and solve for r. Hence 2P = P e r/2) 2 = e r/2) ln2) = r/2) 2 ln2) = r r = 2 ln2) r %) = % ) 2 ln2)

5 Problem 10 v.2 Determine the per annum interest rate required for an investment with continuous compound interest to yield an effective rate of 5.5%. Express your answer as a percent. We know that an investment with continuous compound interest of r yields an amount of money at time t of At) = P e rt and we know that an effective interest rate of 5.5%, yields an amount of money at time t of At) = P ) t = P )t Because these are both the same function, these expressions for the function must be the same for all t, so in particular they are the same for t = 1. So we set P e r1) = P )1) and then we solve for r, the per annum interest rate for the investment with continuous compound interest: P e r1) = P )1) P e r = P ) e r = r = ln ) r = ln ) r %) = % ln )