At the beginning of an experiment, a scientist has 212 grams of radioactive goo. After 120 minutes, her sample has decayed to 6.625 grams.

Question

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

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

How many grams of goo will remain after 35 mins?



Answer 1

Here's a little shorter method.

6.625=212(1/2)^(120/m)

0.03125=(1/2)^(120/m)

ln(0.03125)=ln[(1/2)^(120/m)]

ln(0.03125)=(120/m)ln(0.5)

120/m=ln(0.03125)/ln(0.5)

120/m=-3.465736/-0.693147

120/m=5

120=5m

m=120/5

m=24 minutes for the half-life

second part:A=212(1/2)^(t/24)

A=212(1/2)^(35/24)

A=212(0.5)^1.458

A=212*0.363997

A=77.167364 gm

Answer 2

The formula for decay is:

G(t) = a•(b)^-t where b > 1, or G(t) = a•(b)^t where 0< b < 1

when t = 0, G(t) = 212, therefore

a = 212

and when t = 120, G(t) = 6.625 grams. To find for b:

6.625 = 212(b)^-120. Let see if b>0.

6.625/212 = b^-120

(6.625/212)^-1/120 = (b^-120)^-1/120

b ≈ 1.0293

Therefore the formula is G(t) = 212 (1.0293...)^-t

First question: What is the half-life of the goo in minutes? 

Half life is G(t) = 212 ÷ 2 = 106 grams

106 = 212 (1.0293)^-t

0.5 = (1.0293)^-t

ln (0.5) = ln [(1.0293)^-t]

ln (0.5) = -t ln (1.0293)

ln (0.5) / ln (1.0293) = -t

-24 = -t

t = 24 minutes is the half life of goo.

Second question: Find a formula for G(T) to amount of goo remaining at time t. G(T)=

This has already been solved:

G(t) = 212 (1.0293...)^-t

Third question: How many grams of goo will remain after 35 mins?

G(35) = 212(1.0293)^-35

G(35) ≈ 77.15 grams

(Note: Make sure you use all the decimal numbers in your calculator for 1.0293... and just round off only on your final answer to make it accurate.)