Question 315557
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A.


In general:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ P(t)\ =\ P_oe^{rt}]


For a 3.1% interest rate:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ P(t)\ =\ P_oe^{0.031t}]


B.  1 yr:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ P(1)\ =\ 1000e^{0.031}]


B.  2 yr:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ P(2)\ =\ 1000e^{0.031\,\cdot\,2}]


The calculator work is left as an exercise for the student.


C.  The investment will double regardless of the value of the initial investment when *[tex \LARGE \frac{P(t)}{P_o}\ =\ 2], which is to say when, in general,  *[tex \LARGE e^{rt}\ =\ 2], or specifically for this problem, *[tex \LARGE e^{0.031t}\ =\ 2].


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ e^{rt}\ =\ 2]


Take the natural log of both sides:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \ln\left(e^{rt}\right)\ =\ \ln(2)]


Use *[tex \LARGE \log_b(x^n)\ =\ n\log_b(x)]:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ rt\ln\left(e\right)\ =\ \ln(2)]


Use *[tex \LARGE ln(e)\ =\ 1]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ rt\ =\ \ln(2)]


Divide by *[tex \LARGE r]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ t\ =\ \frac{\ln(2)}{r}]


Is the general solution.  The specific solution and the resulting arithmetic are left as an exercise for the student.


In a still more general case, the time, *[tex \LARGE t], to multiply the principal amount by *[tex \LARGE m], with interest rate, *[tex \LARGE r], compounded continuously is:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ t\ =\ \frac{\ln(m)}{r}]


By the way, this was 4 problems in one post.  You are allowed only 1 problem per post.  Read the instructions, that's why they are there.


John
*[tex \LARGE e^{i\pi} + 1 = 0]
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