The mass of a radioactive substance follows a continuous exponential decay model. A sample of this radioactive substance has an initial mass of 3691 kg and decreases continuously at a relative rate of \( 9 \% \) per day. Find the mass of the sample after six days. Do not round any intermediate computations, and round your answer to the nearest tenth.
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To find the mass of the sample after six days, we can use the formula for continuous exponential decay, which is given by: \[ M(t) = M_0 e^{-kt} \] where: - \( M(t) \) is the mass at time \( t \), - \( M_0 \) is the initial mass, - \( k \) is the decay constant (in this case, \( k = 0.09 \) for a \( 9\% \) decay rate), - \( t \) is the elapsed time (in days), - \( e \) is the base of the natural logarithm (approximately 2.71828). Given: - \( M_0 = 3691 \text{ kg} \), - \( k = 0.09 \), - \( t = 6 \text{ days} \). Plugging in the values, we have: \[ M(6) = 3691 e^{-0.09 \times 6} \] Calculating the exponent: \[ -0.09 \times 6 = -0.54 \] Now, finding \( e^{-0.54} \): Using a calculator, we find: \[ e^{-0.54} \approx 0.582 \] Now substituting back into the mass equation: \[ M(6) = 3691 \times 0.582 \] Calculating this gives: \[ M(6) \approx 2141.862 \] Rounding to the nearest tenth, the mass of the sample after six days is approximately: **2141.9 kg**.