I need assistance to these questions.

Question

1. What must be invested today, to be worth $20,000 in 10 years, if it is compounded yearly at 8%?

2. Write as a sum or difference of simple (single symbol argument) logs: Log [(AB)/(CD)]

3. The half-life of carbon 14 is 5730 years. How old is a bone containing 20% of its original carbon 14?

Answer 1

the first part is a standard exponential growth problem where Final amount = Initial amount x r^t so

20000= A x (1.08)^10

A = 20000/(1.08)^10

A = 9263.87

The second part is a continuous decay problem where

Final amount=Initial amount x e^rt

First we must calculate r(decay constant )

We know that in 5730 years there will only be 1/2 of the initial amount( that is the definition of half life) so

0.5= e^5730r. now take the natural log of both sides so

ln(0.5)= 5730r = -0.000120968

r=ln(0.5)/5730. We will use this value to now calculate the number of years(t) for sample to decay to 20 percent of its initial amount so

(0.2) = e^rt. Again take natural log of both sides to solve for t

ln (0.2)= rt

t= ln0.2/rt

t= -1.6094379/-0.000120968

t= 13304.66 years