The half-life of radium is 1700 years. Assume that the decay rate is proportional to the amount. An initial amount of 5 grams of radium deca to 3 grams in
A) 850 years
B) 1050 years
C) 1150 years
D) 1250 years
E) 1350 years

Question

Quizplus · Accepted Answer

We can use the equation for exponential decay: 
A(t) = A(0) e^(-kt) 
where A(t) is the amount at time t, A(0) is the initial amount, k is the decay constant, and e is the base of natural logarithms. 
We want to find the time at which A(t) = 3 grams, when A(0) = 5 grams. So we have: 
3 = 5 e^(-k(t)) 
Dividing both sides by 5 gives: 
0.6 = e^(-k(t)) 
Taking the natural logarithm of both sides: 
ln(0.6) = -kt 
Solving for t: 
t = -ln(0.6) / k 
The half-life is 1700 years, which means that A(0)/2 = 2.5 grams remains after 1700 years. Therefore, we can solve for k by plugging in A(0) = 5 and A(1700) = 2.5: 
2.5 = 5 e^(-k(1700)) 
0.5 = e^(-k(1700)) 
ln(0.5) = -k(1700) 
k = ln(2) / 1700 ≈ 0.0004082 
Substituting back into the equation for t: 
t = -ln(0.6) / 0.0004082 ≈ 1254.4 years 
Therefore, the best choice is D) 1250 years.

The Half-Life of Radium Is 1700 Years