At the beginning of an experiment, a scientist has 224 grams of radioactive goo.

Question

At the beginning of an experiment, a scientist has 224 grams of radioactive goo. After 105 minutes, her sample has decayed to 7 grams.

What is the half-life of the goo in minutes?

Find a formula for G(t), the amount of goo remaining at time t. G(t)=

How many grams of goo will remain after 17 minutes?

Answer 1

224 = initial amount of goo

112/224 = 1/2 = e^rt where t = half life, r= rate of change

7 = 224e^105r = grams after 105 minutes

7/224 = e^105r = 1/32

take natural logs of both sides

ln(1/32) = 105r

r= ln(1/32)/105 = about -.033

G(t) = 224e^(ln(1/32)t/105) = about 224e^-.033t

half life = t where

1/2 = e^-.033t

take natural logs

ln.5 = -.033t

t = (ln.5)/-.033 = about

21 minutes

after 17 minutes

G(17) = 224e^-.033(17) = about

127.82 grams of goo

Answer 2

7 = 224×(1/2)^105/h

Divide both sides by 224:

(7/224) = (1/2)^105/h

Take the log base (1/2) of both sides:

log _(1/2) [7/224] = 105/h

h = 105 / log _(1/2) [7/224] = 105/5

= 21 minutes, the half-life.

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In 7 minutes,

x = 224 * (1/2)^t/h = 224 * (1/2)^7/21

x = 224 * (1/2)^1/3

x = 177.79 grams, approximately.