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Exponential and Logarithmic Functions
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 $3.7\mathrm{\%}$ 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.
days

Step 1 ：The problem is about finding the half-life of a radioactive substance that follows a continuous exponential decay model, with a decay rate parameter of $3.7\mathrm{\%}$ per day.

Step 2 ：The half-life of a substance under exponential decay can be calculated using the formula $T=\frac{ln(2)}{k}$, where $k$ is the decay rate.

Step 3 ：In this case, the decay rate is given as $3.7\mathrm{\%}$ per day, which needs to be converted to a decimal form for the calculation. So, $k=0.037$.

Step 4 ：We can substitute this value into the formula and calculate the half-life: $T=\frac{ln(2)}{0.037}$.

Step 5 ：By calculating the above expression, we get $T=18.733707582701225$.

Step 6 ：Rounding to the nearest hundredth, we get the final answer: The half-life of the substance is approximately $\boxed{{\displaystyle 18.73}}$ days.