Finding the rate or time in a word problem on continuous exponential... The mass of a radioactive substance follows a continuous exponential decay model, with a decay rate parameter of \( 7,2 \% \) per day, Find the half-life of this substance (that is, the time it takes for one-half the original amount in a given sample of this substance to decay), Note: This is a continuous exponential decay model, Do not round any intermediate computations, and round your answer to the nearest hundredth.

To find the half-life of a substance that decays continuously, we can use the formula for half-life in terms of the decay rate \( r \): \[ t_{1/2} = \frac{\ln(2)}{r} \] In your case, the decay rate \( r \) is \( 0.072 \) (which is \( 7.2\% \) written as a decimal). Substituting this into the formula gives: \[ t_{1/2} = \frac{\ln(2)}{0.072} \] Calculating this will give you the half-life in days. The value of \( \ln(2) \) is approximately \( 0.693147 \), so: \[ t_{1/2} = \frac{0.693147}{0.072} \approx 9.63 \] So, the half-life of the radioactive substance is approximately \( 9.63 \) days. Remember that this method relies on the continuous exponential decay model. Furthermore, when working with continuous exponential decay, always ensure you convert percentage rates into decimal form for calculations. Many people mistakenly use the percentage directly, which results in incorrect values for \( r \). After finding the half-life, it's also a good idea to check your calculations by ensuring that, after one half-life, only half of the original amount remains, affirming the accuracy of your result!