A quantity with an initial value of 920 decays exponentially at a rate of \( 85 \% \) every week. What is the value of the quantity after 21 days, to the nearest hundredth?

To find the value of the quantity after 21 days, we need to calculate the exponential decay of the quantity over time. Given: - Initial value of the quantity = 920 - Decay rate = 85% per week First, we need to convert the decay rate from percentage to a decimal: \[ 85\% = 0.85 \] Next, we need to calculate the number of weeks in 21 days: \[ 21 \text{ days} = \frac{21}{7} \text{ weeks} = 3 \text{ weeks} \] Now, we can calculate the value of the quantity after 3 weeks using the formula for exponential decay: \[ A = P \times (1 - r)^t \] where: - \( A \) is the final value of the quantity - \( P \) is the initial value of the quantity - \( r \) is the decay rate - \( t \) is the time in weeks Substitute the given values into the formula: \[ A = 920 \times (1 - 0.85)^3 \] Now, we can calculate the value of the quantity after 21 days. Calculate the value by following steps: - step0: Calculate: \(920\left(1-0.85\right)^{3}\) - step1: Subtract the numbers: \(920\times 0.15^{3}\) - step2: Convert the expressions: \(920\left(\frac{3}{20}\right)^{3}\) - step3: Evaluate the power: \(920\times \frac{27}{8000}\) - step4: Multiply: \(\frac{621}{200}\) The value of the quantity after 21 days is approximately 3.105. Therefore, the value of the quantity after 21 days, to the nearest hundredth, is 3.11.