a) The given function is expressed as
y = yoe^-0.005t
The rate is - 0.005
Since it is negative, it means that the situation is that of an exponential decay
b) The decay rate is - 0.005
c) The constant, yo represents the initial amount of radioactive isotope
d) When 95% of the sample has decayed, it means that the quantity that has decayed is
95/100 x yo = 0.95yo
This means that when y = 0.95yo, the isotope is no longer useful. To determine the time it will take for 0.95yo to decay, we would substitute y = 0.95yo into the function and solve for t. We have
0.95yo = yoe^-0.005t
0.95yo/ yo = e^-0.005t
0.95 = e^-0.005t
Taking the natural log of both sides of the equation,
ln 0.95 = ln e^-0.005t
By applying the law of logarithms to the right side of the equation,
ln 0.95 = - 0.005t ln e
Recall, ln e = 1
Thus, we have
ln 0.95 = - 0.005t
t = ln 0.95/- 0.005
t = 10.3
It will take 10.3 minutes for 95% of the sample to decay
e) If the student writes the function as
y = yo/e^-0.005t, he is incorrect because the rule of exponents was not applied properly. It should have been
y = yo/e^0.005t