THe nucleus need help Physics

The half-life of 92U against alphy decay is 4.5x10 yr. (a) How long does it take for seven-eigths of a sample of this isotope to decay? (b) For fifteen-sixteenths to decay

This is more of a math problem than a physics problem. You want to use the equation of N=No*e^(k*t) where k is the rate of decay, N is the final amount, No is the initial amount, and t is time.

So to solve for k, you can plug in a set of variables you know. Since you know the half life, that means that half of the sample will decay in a given time (1 year).

So for simplicity, you can use numbers like 5 and 10. So 5=10*e^(k*1) (<- Saying time is in years).

1/2=e^(k)
=>ln(1/2)=k

So to solve for seven eights of the sample, 7/8=1*e^(ln(1/2)*t) (<- Time is the unknown (in years) of the final sample being 7/8 of the initial sample 1)

7/8=e^(ln(1/2)*t)
ln(7/8)=ln(1/2)*t
ln(7/8)/ln(1/2)=t

So that'll be the number of years :) You can then convert to days/hours/etc from that number of years!

a) for this 3 half lives are required, so the time required is t = 3 x45 = 135 years

b) for this 4 half lives are required so, the time required is t = 4 x45 = 180 years.