To determine the half-life of a substance, we need to find the time it takes for the amount of the substance to decrease by half.
In this case, the percent of the material remaining after t years is given by the equation P(t) = 100(1.2)^-t.
To find the half-life, we need to find the value of t for which P(t) is equal to 50%.
So, we can set up the equation:
50 = 100(1.2)^-t
To solve for t, we can start by dividing both sides of the equation by 100:
0.5 = (1.2)^-t
Next, we can take the logarithm of both sides of the equation. The choice of logarithm base doesn't matter, but let's use the natural logarithm (ln) for this example:
ln(0.5) = ln((1.2)^-t)
Using the property of logarithm that states ln(a^b) = b * ln(a), we can simplify further:
ln(0.5) = -t * ln(1.2)
Now, we can solve for t by dividing both sides of the equation by -ln(1.2):
t = ln(0.5) / -ln(1.2)
Using a calculator or computer software to evaluate this expression, we find that t is approximately 9.08 years.
Therefore, the half-life of the substance is approximately 9.08 years.