MATH 147
Quiz Key #2
2/4/2016
(1) Polonium 210 Po210 has a half-life of 140 days.
(a) If a sample of Po210 has an initial mass of 300 micrograms, find a formula
for the mass after t days.
(b) How long would it take this sample to decay to 20% of the original
amount?
Solution: (a) We look for a formula of the form y (t) = y0 ekt . Then y (140) =
1
2 (300) = 150 and y0 = 300.
150 = 300e140k .
1
∴ = e140k .
2
1
= 140k.
∴ ln
2
1
∴k=−
ln 2.
140
Hence, y (t) = 300 · e−(t/140) ln 2 .
(b) Now using this formula and the fact that 20% of 300 micrograms is 60
micrograms, we get that
60 = 300 · e−(t/140) ln 2 .
1
∴ = e−(t/140) ln 2 .
5
t · ln 2
∴ − ln 5 = −
.
140
Therefore, t =
140 ln 5
ln 2
days.
(2) Use the given semilog plot to find the functional relationship between y
and x.
Solution: After converting to semilogarithmic coordinates ( x, log y) , we have
a line passing through the points (0, 1) and (3, 3) . The slope of this line is given
−1 = 2 . Therefore,
by m = 33−
0
3
log y =
2
x + 1.
3
∴ y = 10(2/3)x+1 .
1