What is the figure that gives the half-life of the isotope in minutes to 4 significant figures?
Exponential regression on a set of data, which gives the amount of atoms of a radioactive isotope remaining after a time X minutes, produced the following output.
ExpReg
y = a*b^x
a = 10001
b = 0.90002
To find the half-life of the isotope in minutes, we need to consider the equation generated by the exponential regression output. The equation is:
y = a * b^x
From the equation, we can see that 'a' represents the initial amount of atoms present, and 'b' represents the decay factor or the rate at which the isotope decays over time.
To find the half-life, we need to find the time it takes for the initial amount of atoms to reduce by half, which means finding the value of 'x' when 'y' is equal to half of the initial amount.
Let's calculate it step by step:
1. Find the half-life equation:
Half of the initial amount is given by a/2, so we set y = (a/2).
Therefore, (a/2) = a * b^x.
2. Solve for 'x':
Divide both sides of the equation by 'a' to isolate b:
(a/2a) = (a * b^x) / a
1/2 = b^x
Now, we need to solve for 'x'. Taking the logarithm of both sides can help us do that.
3. Take the logarithm of both sides:
log(1/2) = log(b^x)
log(1/2) = x * log(b)
4. Solve for 'x':
Divide both sides by log(b):
x = log(1/2) / log(b)
5. Calculate the value of 'x':
Using the given values of a and b:
x = log(1/2) / log(0.90002)
To find the half-life in minutes, we need to put this value of 'x' into the equation. However, please note that the given values are only rounded to five decimal places and might not provide the desired level of accuracy for calculating the half-life. It would be better to use the original, more precise values for calculations.