A quantity with an initial value of 7200 decays exponentially at a rate of \( 0.3 \% \) every day. What is the value of the quantity after 10 weeks, to the nearest hundredth?

To find the value of the quantity after 10 weeks, we first need to determine how many days that is. Since there are 7 days in a week, 10 weeks equals \( 10 \times 7 = 70 \) days. The formula for exponential decay is given by: \[ A = A_0 e^{-kt} \] where: - \( A_0 \) is the initial amount (7200), - \( k \) is the decay constant (in this case, \( 0.3\% \) per day, or \( 0.003 \) in decimal form), - \( t \) is the time in days. Plugging in the values, \[ A = 7200 e^{-0.003 \times 70} \] Calculating the exponent: \[ -0.003 \times 70 = -0.21 \] Now substituting into the equation: \[ A = 7200 e^{-0.21} \] Calculating \( e^{-0.21} \) (approximately \( 0.8107 \)): \[ A \approx 7200 \times 0.8107 \approx 5845.04 \] Thus, the value of the quantity after 10 weeks is approximately **5845.04**.