Final answer:
To determine when only 30% of a radioactive substance with a 5.9% annual decay rate remains, we solve the exponential decay equation and find that it takes approximately 20.4 years.
Step-by-step explanation:
To calculate how long it will take for only 30% of a radioactive substance to remain given a continuous decay rate of 5.9% per year, we use the formula for exponential decay:
A = P * e^(-rt)
Where:
- A = the amount of substance remaining after time t
- P = the initial amount of the substance
- r = the decay rate per unit of time
- t = the time that has passed
- e = the base of natural logarithms (approximately 2.71828)
In this scenario, we want to find the time t when A is 30% of P. We set up the equation:
0.30P = P * e^(-0.059t)
To solve for t, we divide both sides by P and take the natural logarithm of both sides:
ln(0.30) = ln(e^(-0.059t))
-1.20397 = -0.059t
Finally, we divide both sides by -0.059 to solve for t:
t = 20.404
Therefore, it will take approximately 20.4 years for 30% of the original amount of the substance to remain.