Question

How do you determine the exponential function for a half-life problem?

Answer 1

Hint: This type of problem is based on the concept of applications of exponential function. First, let us assume the half-life of a radioactive substance be x. Consider y to be the amount of the substance initially. To find the quantity after t years, we have to do necessary calculations. Divide the t by x which will be the power of \[\dfrac{1}{2}\] that is \[{{\left( \dfrac{1}{2} \right)}^{\dfrac{t}{x}}}\]. Then, multiply the obtained exponential value with the initial quantity y. thus, we get the quantity function of the radioactive substance after t years.

Complete step-by-step solution:
According to the question, we are asked to determine the exponential function for a half-life problem.
We have to consider a radioactive substance with known half-life h.
Let us assume that the half-life of the considered radioactive substance be x that is h=x.
We have been given initially some quantity of the radioactive substance. Let us say the quantity be y grams.
Therefore, we can express as Q(0)=y
Now, we have to find the quantity function of the radioactive substances after some years, say t years.
Therefore, we have to find Q(t).
We know that the formula to find the quantity function of a radioactive substance is
\[Q\left( t \right)=Q\left( 0 \right){{\left( \dfrac{1}{2} \right)}^{\dfrac{time}{period}}}\]
Here, \[\dfrac{1}{2}\] is considered because it is a half-life problem.
The time in the formula refers to the years we have considered that is t.
And the period is the half-life of the substance that is x.
Therefore, we get
\[Q\left( t \right)=Q\left( 0 \right){{\left( \dfrac{1}{2} \right)}^{\dfrac{t}{x}}}\]
Here, we know that the initial amount considered is equal to y.
Therefore, we get
\[Q\left( t \right)=y{{\left( \dfrac{1}{2} \right)}^{\dfrac{t}{x}}}\]
Hence, the half-life problems can be solved by using exponential function with the formula \[Q\left( t \right)=Q\left( 0 \right){{\left( \dfrac{1}{2} \right)}^{\dfrac{time}{period}}}\].

Note: For this type of problem, we should consider \[\dfrac{1}{2}\] since the problem is half-life. The growth or decay varies exponentially. We should be very careful in finding the exponential values. Avoid calculation mistakes based on sign conventions.