If the rate, r, at which people get sick during an epidemic of flu is approximately $r=500 t e^{-0.5 t}$ , where r is people per day and t is time in days since the start of the epidemic, then the total number of people who get sick is 2000.How does the total number of people getting sick change if the exponent is tripled and the rate is now $q=500 t e^{-1.5 t}$ ?
A)The number is one-third as large..
B)The number is one-ninth as large.
C)The number is three times as large.
D)The number is nine times as large.

Question

Quizplus · Accepted Answer

The total number of people getting sick is given by the integral of the rate function over time. Tripling the exponent in the exponential function makes the rate decrease faster as time increases, leading to a smaller total number of people getting sick. Specifically, the integral of $e^{-1.5t}$ compared to $e^{-0.5t}$ over any given time period will be significantly smaller, but not simply by a factor of 3. The effect of the exponential function is not linear, and the change in the exponent has a more pronounced effect, leading to the total number being one-ninth as large due to the square of the tripling factor (since the integral involves the reciprocal of the exponent, and the area under the curve is significantly reduced).

If the Rate, R, at Which People Get Sick During r=500te−0.5tr=500 t e^{-0.5 t}r=500te−0.5t

If the Rate, R, at Which People Get Sick During $r=500 t e^{-0.5 t}$