16. A technician places 50 mg of a radioactive substance into a laboratory chamber.
After 10 minutes, 41 mg remain.
To calculate the half-life of the radioactive substance, we can use the formula:
Final amount = Initial amount * (1/2)^(time/half-life)
In this case, the initial amount is 50 mg, and the final amount is 41 mg. The time is 10 minutes. We can rearrange the formula to solve for the half-life:
(4/5) = (1/2)^(10/half-life)
To simplify this equation, we can take the logarithm of both sides (base 1/2):
log(4/5) = (10/half-life) * log(1/2)
Using a calculator, we find that log(4/5) = -0.10491 and log(1/2) = -0.30103. Substituting these values into the equation, we get:
-0.10491 = (10/half-life) * (-0.30103)
Now we can isolate the half-life:
0.10491 = (10/half-life) * 0.30103
Dividing both sides by 0.30103:
0.34781 = 10/half-life
Multiply both sides by half-life:
0.34781 * half-life = 10
Divide both sides by 0.34781:
half-life = 10 / 0.34781
Using a calculator, we find that the half-life is approximately 28.748 minutes.