I have found the answer on Chegg, and have attached a picture below. But I just

Question

I have found the answer on Chegg, and have attached a picture below. But I just wondered if someone could explain the points at which the equation is rearranged and where all the numbers in the answer come from.
The formula I have is t=1/lambda * ln(No/N) but the image below dives straight into rearrangement without saying why etc. Thanks! 0.693 Rate constant, K 0.693 1.27x10 yr 5.456 x100 yr 05.46 x10- yr To calculate the age of the rock 5.4 x 10 yr (4.2+1)J -1.6486 (5.46x10 10 yr i)t 1.6486 (546x10 yr-1) 1.6486 46 x10 yr 30x109

Explanation / Answer

t=(1/lambda)*ln(N0/N)

where N0=initial amount of 40K

N=current amount of 40K

now inititally there was no 40Ar and only 40K.

so current amount of 40Ar=N0-N

given (N0-N)/N=4.2

==>N0-N=4.2*N

==>N0=5.2*N

lambda=0.693/half life=5.4567*10^(-10)

then t=(1/(5.4567*10^(-10)))*ln(5.2)

=3.0213*10^9 years

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