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QUESTION

# What is the rock's age? A rock contains one-fourth of its original amount of potassium-40 The half-life of potassium-40 is 1.3 billion years?

The rock's age is approximately "2.6 billion" years.

There are essentially two ways of solving problems. One way is by applying the half-life formula, which is

A(t) = A_0(t) * (1/2)^(t/t_(1/2)) , where

A(t) - the quantity that remains and has not yet decayed after a time t; A_0(t) - the initial quantity of the substance that will decay; t_(1/2) - the half-life of the decaying quantity;

In this case, the rock contains "1/4th" of the orignal amount of potassium-40, which means A(t) will be equal to (A_0(t))/4. Plug this into the equation above and you'll get

(A_0(t))/4 = A_0(t) * (1/2)^(t/t_(1/2)), or 1/4 = (1/2)^(t/t_(1/2))

This means that t/t_(1/2) = 2, since 1/4 = (1/2)^2.

Therefore,

t = 2 * t_(1/2) = 2 * "1.3 = 2.6 billion years"

A quicker way to solve this problem is by recognizing that the initial amount of the substance you have is halved with the passing of each half-life, or t_(1/2).

This means that you'll get

A = (A_0)/2 after the first 1.3 billion years

A = (A_0)/4 after another 1.3 billion years, or 2 * "1.3 billion"

A = (A_0)/8 after another 1.3 billion years, or 2 * (2 * "1.3 billion")

and so on...