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. days

In a continuous exponential decay model, the mass as a function of time is given by   m(t) = m₀ · e^(–kt), where m₀ is the initial mass and k is the decay rate. The half-life, t₁/₂, is the time it takes for the mass to become half of m₀, so we have   m₀/2 = m₀ · e^(–k · t₁/₂). Dividing both sides by m₀:   1/2 = e^(–k · t₁/₂). Taking the natural logarithm of both sides:   ln(1/2) = –k · t₁/₂. Recall that ln(1/2) = –ln2, so:   –ln2 = –k · t₁/₂  ⟹  t₁/₂ = ln2 / k. Plug in k = 0.072 (since 7.2% per day = 0.072 per day):   t₁/₂ = ln2 / 0.072. Now, compute t₁/₂:   ln2 ≈ 0.6931471805599453,   t₁/₂ ≈ 0.6931471805599453 / 0.072 ≈ 9.626 (days). Rounded to the nearest hundredth:   t₁/₂ ≈ 9.63 days. Thus, the half-life of the substance is approximately 9.63 days.