Question

The formula for continuous compound interest is A = Pert, where A is the amount of money after compounding, P is the original amount invested, r is the annual interest rate expressed as a decimal, and t is the number of years. For which of these functions does T(r) represent the number of years it would take an amount of money to triple if it were compounded continuously at r percent per year? A) T(r) = r ln 3 B) T(r) = ln 3 r C) T(r) = ln r 3 D) T(r) = 3 ln r

Answer 1

Answer:

Option B. $T(r)=\frac{ln(3)}{r}$

Step-by-step explanation:

we know that

The formula to calculate continuously compounded interest is equal to

$A=P(e)^{rt}$  

where  

A is the Final Investment Value  

P is the Principal amount of money to be invested  

r is the rate of interest in decimal  

t is Number of Time Periods  

e is the mathematical constant number

Let

x-------> the Principal amount of money to be invested

we have  

$A=\ 3x\\ P=\ x\\ r=r$  

substitute in the formula above  and solve fot t

$\ 3x=\ x(e)^{rt}$  

$3=(e)^{rt}$  

Applying ln both sides

$ln(3)=rt*ln(e)\\ \\rt=ln(3)\\ \\t=\frac{ln(3)}{r}$

Convert to function notation

$T(r)=\frac{ln(3)}{r}$

Answer 2

Answer:

T(r) = ln 3 /r

Step-by-step explanation:

The formula for continuous compound interest is A = Pert, so if both sides of the equation are divided by P, it becomes A/P= ert. Since A/P = 3, the equation becomes 3 = ert, and this equation written in logarithmic form is ln 3 = rt. If both sides are divided by r, the equation becomes t = ln 3 /r, and this equation written as the function T in terms of r is T(r) = ln 3 /r