Questions: After the creation and implementation of a vaccine, the number of people that catch a disease falls by 40% every 12 days. Given this, compute the per-day rate of decay of the number of people that catch the disease. Give as a percentage, without the percent symbol, accurate to two decimal places (for example, if your answer is 12.6821%, you should input 12.68).

Solution Steps

To find the per-day rate of decay, we need to determine the daily decay rate that results in a 40% reduction over 12 days. This can be modeled using exponential decay. We can use the formula for exponential decay, where the final amount is the initial amount times (1 minus the decay rate) raised to the power of the number of days. We solve for the daily decay rate.

We need to find the per-day rate of decay that results in a 40% reduction over 12 days. This can be modeled using exponential decay.

The exponential decay formula is given by: \[ N(t) = N_0 \times (1 - r)^t \] where \( N(t) \) is the final amount, \( N_0 \) is the initial amount, \( r \) is the daily decay rate, and \( t \) is the time in days. We know that \( N(t) = 0.6 \times N_0 \) after 12 days, so: \[ 0.6 = (1 - r)^{12} \]

To find \( r \), we take the 12th root of both sides: \[ 1 - r = 0.6^{\frac{1}{12}} \] Thus, the daily decay rate \( r \) is: \[ r = 1 - 0.6^{\frac{1}{12}} \]

Using the calculation, we find: \[ r \approx 0.04168 \] Converting this to a percentage: \[ r \times 100 \approx 4.168\% \]

Final Answer