The equation for the half life is expressed as
y = a(0.5)^t/h
where
y represents the final amount after t minutes
a represents the initial amount
0.5 is the decay rate which is half
t is the time
h is the time taken for half of the element to decay.
From the information given,
a = 900
y = 121
h = 6
We want to find t.
By substituting the values into the equation, we have
121 = 900(0.5)^t/6
Dividing both sides of the equation by 900, we have
121/900 = 900/900(0.5)^t/6
0.134 = 0.5^0.17t
Taking the natural log of both sides of the equation, we have
ln 0.134 = ln 0.5^0.17t
On the rightm we would apply one of the rules of logarithm which is expressed as
lna^x = xlna
Thus, we have
ln 0.134 = 0.17t ln0.5
Dividing both sides of the equation by ln 0.5, we have
ln 0.134/ln 0.5 = 0.17t ln0.5/ln 0.5
0.17t = 2.8997
t = 2.8997/0.17
t = 17.1
The time it will take the element to decay to 121 grams is 17.1 minutes