Question

Suppose you find a rock that contains some potassium-40 (half-life of 1.3 billion years). you measure the amount and determine that there are 5 grams of potassium-40 in the rock. by measuring the amount of its decay product (argon-40) present in the rock, you realize that there must have been 40 grams of potassium-40 when the rock solidified. how old is the rock?

Answer 1

Equation for Half life :
A = a(0.5)^(t/h)
A is current amount, "a" is initial amount, h is halflife, t is time

5 = 40(0.5)^(t/1.3x10^9)
5/40 = (0.5)^(t/1.3x10^9)
take the log of both sides , power rule
Log(5/40) = (t/1.3x10^9) * Log(0.5)
(1.3x10^9) * Log(5/40) / Log(0.5) = t
3.9x10^9 years = t

And if you think about what a half life is, the time it take for the amount to reduce to half.
40/2 = 20
20/2 = 10
10/2 = 5
It went through 3 half-lifes
3 * 1.3x10^9 = 3.9x10^9 years