A radioactive substance decays according to the formula
Q(t) = Q0e−kt
where Q(t) denotes the amount of the substance present at time t (measured in years), Q0 denotes the amount of the substance present initially, and k (a positive constant) is the decay constant.
(a) Find the half-life of the substance in terms of k.
(b) Suppose a radioactive substance decays according to the formula
Q(t) = 36e−0.0001074t
How long will it take for the substance to decay to half the original amount? (Round your answer to the nearest whole number.)
(a) The half-life of the substance is given by t1/2 = ln(2)/k.
(b) The amount of the substance present at time t is given by Q(t) = 36e−0.0001074t. To find the time it takes for the substance to decay to half the original amount, we solve for t when Q(t) = 18. This gives us t = ln(2)/0.0001074 ≈ 6,400 years.
(a) To find the half-life of the substance, we need to determine the time it takes for the amount to reduce to half of its initial value (Q0/2).
Given Q(t) = Q0e^(-kt), we can set Q(t) = Q0/2 and solve for t.
Q(t) = Q0e^(-kt)
Q0/2 = Q0e^(-kt)
Dividing both sides by Q0, we get:
1/2 = e^(-kt)
Taking the natural logarithm of both sides:
ln(1/2) = ln(e^(-kt))
ln(1/2) = -kt * ln(e)
ln(1/2) = -kt
Now, we need to solve for t. Recall that ln(1/2) is the natural logarithm of 1/2, which is equal to -0.6931. Also, ln(e) is equal to 1.
-0.6931 = -kt
Dividing both sides by -k:
t = 0.6931/k
Therefore, the half-life of the substance is 0.6931/k.
(b) In this case, we have the equation:
Q(t) = 36e^(-0.0001074t)
To find the time it takes for the substance to decay to half its original amount, we need to solve for t when Q(t) = Q0/2.
Q(t) = 36e^(-0.0001074t)
36/2 = 36e^(-0.0001074t)
Dividing both sides by 36:
1/2 = e^(-0.0001074t)
Taking the natural logarithm of both sides:
ln(1/2) = ln(e^(-0.0001074t))
ln(1/2) = -0.0001074t * ln(e)
ln(1/2) = -0.0001074t
Using ln(1/2) = -0.6931, we have:
-0.6931 = -0.0001074t
Dividing both sides by -0.0001074:
t = -0.6931/-0.0001074
Simplifying:
t ≈ 6,444.19
Rounding to the nearest whole number, it will take approximately 6,444 years for the substance to decay to half its original amount.
To find the half-life of the substance in terms of k, we need to solve for the time at which Q(t) is half of the initial amount Q0.
(a) Half-life in terms of k:
Setting Q(t) = Q0/2 in the decay formula:
Q(t) = Q0e^(-kt) = (Q0/2)
Dividing both sides of the equation by Q0:
e^(-kt) = 1/2
Next, take the natural logarithm (ln) of both sides to solve for t:
ln(e^(-kt)) = ln(1/2)
Using the logarithmic property of ln(x^a) = a ln(x):
-kt = ln(1/2)
Finally, solving for t gives:
t = -ln(1/2) / k
The half-life of the substance in terms of k is t = -ln(1/2) / k.
(b) To find the time it takes for the substance to decay to half its original amount in the given formula Q(t) = 36e^-0.0001074t, we need to set Q(t) equal to half of the initial amount.
Q(t) = 36e^-0.0001074t = 36 / 2 = 18
Now we can solve for t:
18 = 36e^-0.0001074t
Dividing both sides by 36:
e^-0.0001074t = 1/2
Taking the natural logarithm (ln) of both sides:
-0.0001074t = ln(1/2)
Solving for t:
t = ln(1/2) / -0.0001074
Calculating the expression on the right-hand side gives us approximately -6,440.62.
Rounding to the nearest whole number, it will take approximately 6,441 years for the substance to decay to half its original amount.