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.

To solve the problem of finding the mass of a radioactive substance after six days, we can use the continuous exponential decay formula: \[ M(t) = M_0 e^{-kt} \] where: - \( M(t) \) is the mass at time \( t \), - \( M_0 \) is the initial mass, - \( k \) is the decay constant (relative rate of decay), - \( t \) is the time in days, - \( e \) is the base of the natural logarithm. ### Step 1: Identify the known values - Initial mass, \( M_0 = 3691 \) kg - Relative rate of decay, \( k = 0.09 \) (since \( 9\% = 0.09 \)) - Time, \( t = 6 \) days ### Step 2: Substitute the values into the formula We need to calculate \( M(6) \): \[ M(6) = 3691 e^{-0.09 \times 6} \] ### Step 3: Calculate the exponent First, calculate the exponent: \[ -0.09 \times 6 = -0.54 \] ### Step 4: Calculate \( e^{-0.54} \) Now we need to calculate \( e^{-0.54} \): \[ e^{-0.54} \approx 0.582 \] ### Step 5: Calculate the mass after 6 days Now substitute back into the mass formula: \[ M(6) = 3691 \times 0.582 \] ### Step 6: Perform the multiplication Now we calculate: \[ M(6) = 3691 \times 0.582 \] Let's calculate this value. Calculate the value by following steps: - step0: Calculate: \(3691\times 0.582\) - step1: Multiply the numbers: \(2148.162\) The calculation gives us: \[ M(6) \approx 2148.162 \text{ kg} \] ### Step 7: Round the result Rounding this to the nearest tenth, we get: \[ M(6) \approx 2148.2 \text{ kg} \] ### Final Answer The mass of the sample after six days is approximately **2148.2 kg**.