MATH SOLVE

4 months ago

Q:
# 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.

Accepted Solution

A:

Answer:t = 21.97 yearsStep-by-step 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)!!!