Question

HELPPPPP!!!!

An investment in a savings account grows to three times the initial value after t years.
If the rate of interest is 5%, compounded continuously, t = years.

Answer 1

t = 22 years

Explanation:

* Lets explain the compound continuous interest

- Compound continuous interest can be calculated using the formula:

A = P e^rt

# A = the future value of the investment, including interest

# P = the principal investment amount (the initial amount)

# r = the interest rate

# t = the time the money is invested for

- The formula gives you the future value of an investment,

which is compound continuous interest plus the

principal.

* Now lets solve the problem

∵ The initial investment amount is P

∵ The future amount after t years is three times the initial value

∴ A = 3P

∵ The rate of interest is 5%

∴ r = 5/100 = 0.05

- Lets use the rule above to find t

∵ A = P e^rt

∴ 3P = P e^(0.05t)

- Divide both sides by P

∴ 3 = e^(0.05t)

- Insert ㏑ for both sides

∴ ㏑(3) = ㏑(e^0.05t)

- Remember ㏑(e^n) = n ㏑(e) and ㏑(e) = 1, then ㏑(e^n) = n

∴ ㏑(3) = 0.05t

- Divide both sides by 0.05

∴ t = ㏑(3)/0.05 = 21.97 ≅ 22

* t = 22 years

Answer 2

t = 21.97 years

Explanation:

The formula for the continuous compounding if given by:

A = p*e^(rt); where A is the amount after compounding, p is the principle, e is the mathematical constant (2.718281), r is the rate of interest, and t is the time in years.

It is given that p = $x, r = 5%, and A = $3x. In this part, t is unknown. Therefore: 3x = x*e^(0.05t). This implies 3 = e^(0.05t). Taking natural logarithm on both sides yields ln(3) = ln(e^(0.05t)). A logarithmic property is that the power of the logarithmic expression can be shifted on the left side of the whole expression, thus multiplying it with the expression. Therefore, ln(3) = 0.05t*ln(e). Since ln(e) = 1, and making t the subject gives t = ln(3)/0.05. This means that t = 21.97 years (rounded to the nearest 2 decimal places)!!!