A radioactive substance decays exponentially. A scientist begins with 140 milligrams of a radioactive substance. After 21 hours, 70 mg of the substance remains. How many milligrams will remain after 35 hours?
To determine the amount of radioactive substance remaining after a certain time, we can use the exponential decay formula. The formula is given by:
A(t) = A₀ * e^(-kt)
Where:
A(t) is the amount of the substance remaining after time t,
A₀ is the initial amount of the substance,
k is the decay constant, and
e is Euler's number (approximately 2.71828).
We are given the initial amount, A₀ = 140 mg, and the amount remaining after 21 hours, A(21) = 70 mg. We can use this information to find the decay constant, k.
Substituting the values into the formula:
70 = 140 * e^(-21k)
Dividing both sides by 140:
0.5 = e^(-21k)
Taking the natural logarithm of both sides to isolate the exponent:
ln(0.5) = -21k
Using a calculator, we find that ln(0.5) ≈ -0.6931.
Therefore,
-0.6931 = -21k
Dividing both sides by -21:
k ≈ 0.033
Now that we have the decay constant, we can use it to find the amount of substance remaining after 35 hours, A(35).
A(35) = 140 * e^(-0.033 * 35)
Using a calculator, we get:
A(35) ≈ 50.9 mg
Therefore, approximately 50.9 milligrams of the substance will remain after 35 hours.