Question

A certain element has a half life of 4.5 billion years.

(a). You find a rock containing a mixture of the element and lead. You determine that 35% of the original element remains; the other 65% decayed into lead. How old is the rock?
(b). Analysis of another rock shows that it contains 55% of it's original element; the other 45% decayed into lead. How old is the rock.

Answer 1

A ) 50% is gone so 4.5 billion right of the bat. Then another 4.5 if it got down to 25% but it’s still at 35%. So it lost 15% from 50%. 15/25 * 4.5 billion = 2.7 billion +the original 4.5 = 7.2 billion years old.

B) it loses 45%. 50% would be 4.5 billion. 45/50 * 4.5 billion = 4.05 billion years old

Answer 2

Explanation:

Given that a certain element has a half life of 4.5 billion years.

For half life of 4.5 billion years we have the equation as

`P(t) = P_0(t)((1)/(2) )^{(t)/(t_(1/2) ) }`

=
`P_0(t)((1)/(2) )^{(t)/(4.5) } }` where t is in billions of years.

When P(t) = 35% of original we have

`P(t) = 0.35 P_0(t) = P_0(t)((1)/(2) )^{(t)/(t_(1/2) ) }\\0.35 = ((1)/(2) )^{(t)/(4.5) } }\\ln0.35 = {(t)/(4.5) }ln ((1)/(2) \\t =6.816`

After 6.816 billion years.

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b) Here
`P(t) = 0.55 P_o(t)`

`t = (log 0.55)/(log 0.5) (4.5)\\t=3.88`

After 3.88 billion years.