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Suppose you invest a dollars to earn an annual interest rate of r percent (as a decimal). After t years, the value of the investment with interest compounded yearly is A(t)=a(1+r) t . The value with interest compounded continuously is A(t)=a . e rt
c. For each situation find the unknown quantity, such that continuous compounding gives you an $ 1 advantage over annually compounded interest. Show your work.
- How much must you invest for 1 year at 2 % ?
- At what interest rate must you invest $ 1000 for 1 year?
- For how long must you invest $ 1000 at 2 % ?

Respuesta :

The sum a = 4966.7 is invested at a rate of r = 4.439% for a period of t=4.621 years.

What exactly is a continuous compounding formula?

  • The continuous compounding formula should be used when an issue expressly states that the amount is "constantly compounded."
  • This formula makes use of the mathematical constant "e," which has a value of approximately 2.7182818.

The continuous compounding formula is as follows:

  • A = Pe^rt
  • Where P represents the starting sum, A represents the total sum, r represents the interest rate, t represents time, and e is a mathematical constant.

So,

According to the first point, we know that atr = 0.02,  and t = 1.

  • a(e^rt) - a(1+r)^t
  • = a[ e^(0.02) - 1.02]
  • = 0.00020134 a

Thus for advantage > 1$ we need is as follows:

  • 0.00020134 a > 1
  • a > 1/ 0.00020134
  • a~4966.7

According to the second point, we got that:

  • t1= 1,
  • a= 1000

Then,

  • 1000[e^r-1-r]> 1
  • >> e^r-1-r > 0.001
  • >>e^r-1-r - 0.001 > 0 ......(1)

Now we hold that r > 0.04439:

  • r~ 0.04439
  • r~ 4.439%

According to the third point, we got to know that:

  • a = 1000,
  • r = 0.02,
  • 1000[e^0.02t - (1.02)^t] > 1
  • >>e^0.02t - (1.02)^t > 1 ......(2)

Now 2nd equation holds t > 4.621.

Therefore, the sum a = 4966.7 is invested at a rate of r = 4.439% for a period of t=4.621 years.

Know more about continuous compounding formula here:

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