Question

What is the half-life of a Radium-226 if its decay rate is $0.000436$ ?
A) $t = 1237$
B) $t = 1365$
C) $t = 1440$
D) $t = 1590$

Answer 1

The given decay rate of Radium-226 we will use the growth function to find the general equation of the half-life of the radium-226. Then we will substitute the given values in the formula and rearrange the terms so that we can simplify it to our advantage.

Complete step by step answer:
The given decay rate of the Radium-226 is $0.000436$ .
That means Radium is decaying at this rate.
We need to find the amount of time it will take to become half of its half-life of the Radium-226.
We know that the exponential growth function is given by the following:
$A = {A_0}{e^{ - rt}}$ … (1)
In the above formula first understand what each term indicates.
The term $A$ is the mass of the radium present at time $t$ and $r$ is the decay rate.
This implies that ${A_0}$ indicates the amount of Radium-226 present at time $t = 0$.
Therefore, we need to find the time at which the amount is half that means we need to find $t$ for which $A = \dfrac{1}{2}{A_0}$ .
Substitute this in the equation (1).
$\dfrac{1}{2}{A_0} = {A_0}{e^{ - rt}}$
We can cancel ${A_0}$ from both sides and substitute $r = 0.000436$ .
Therefore,
$\dfrac{1}{2} = {e^{0.000436t}}$
Since the variable is in the exponent, we will take logarithm on both sides.
$\ln \left( {\dfrac{1}{2}} \right) = - 0.000436t$
Note that $\ln \left( {\dfrac{1}{2}} \right) = - 0.6391$ .
Simplify the above equation for $t$ as follows:
$t = \dfrac{1}{{0.000436}} \times 0.6391$
This gives us an approximate answer as follows:
$t \approx 1590$
Thus, the correct answer is D.

Note:
Note that the given data is about the radioactive element Radium-226. The decay rate is given so we will use the decay rate and the exponential growth function to calculate the half-life. There are two types of logarithmic: one is $\log$ and the other is $\ln$, so correct logarithmic is used to determine the correct solution.