The force of interest 8(t) is a function of time, and at any time t, measured in years, is given by theformula:(i)(ii)8(t) = {0.30.05(0.3 -0.08t 0 ≤ t < 55 ≤tDerive, and simplify as far as possible, expressions for the accumulated amount at (any) timet of a unit investment made at time t = 0. You should derive separate expressions for eachtime interval.(a) Hence, or otherwise, calculate the accumulation at the end of 10 years of aninvestment of £250 made at time t = 0.(b) Calculate the constant nominal annual interest rate convertible monthly equivalentto the transaction in part (a).(iii)Calculate the present value of a continuous payment stream invested from time t = 6 tot = 10 at a rate of p(t) = 30e0.15+0.15t

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Answered: The force of interest 8(t) is a…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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With t in years, the formulas for balances of two bank accounts, in dollars, are: f(t) = 1100(1.05)t and g(t) = 1500e0.05t(a) Describe the balance of the account modeled by f.(b) Describe the balance of the account modeled by g. State the effective annual yield.(c) What continuous rate has the same effective growth rate as the rate at which f(t) grows?
A young person with no initial capital invests k dollars per year at an annual rate of return r. Assume that investments are made continuously and that the return is compounded continuously. a.Determine the sum S(t) accumulated at any time t.
(a) Find the value of Q when K = 81 and L = 16.(b) What happens to the value of Q if we double both the labour and thecapital? (Call the new value Q˜ and write it in terms of the original one).(c) Find an expression for ln Q in terms of ln K and ln L.
2. A population of deer is increasing at a rate of 5% a year. At t = 0 years, there are 98 deer in the population. (a) Using the information above, write a formula for the deer's population P = f(t). (b) Using your formula from (a), determine the approximate number of deer after 5 years have passed.
Match the formula with the appropriate amount of times compounded per year. A(t) = P(1+r)t Annually Monthly Weekly Quarterly Continuously 1. 2. 3. 4t A(t) = P(1 + ² ) ¹² 4 12t A(t) = P(1 + ₂ ) ¹²² 12 A(t) = P(1+2) 52t 52 5. A(t) = Pert
A savings account with an interest rate r, which is compounded n times per year, and begins with P as the principal (initial amount), has the discrete nt compounding formula A (t) = P(1+)". This is n because we multiply the amount by itself plus a small amount, determined by the interest rate, and the account grows each time the compounding occurs. For continuous compounding, we use the formula A (t) = Pert , and if we have seen this formula before, we may not have gotten a satisfactory answer as to why we use it, other than some vague notion of "compounding infinity times per year". In this exercise, we'll use Bernoulli's Rule to find the connection. It might be helpful to review the "Indeterminate Powers" section of the video before beginning. Why can we write nt lim,→00 P(1+ )"t P limn¬∞ (1+)™ ? n
A client’s orders are growing consistently at x% per year. A) If three years from now orders are 50% higher than today, what is the annual growth rate (x)? B) Write an expression (in terms of x) for how many years it will take until orders are double what they are currently

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