Question

the number of bacteria in a culture increases rapidly. the table below gives the number n(t) of bacteria at a few times t (in hours) after the moment when n = 1000.
time t (hours) 0 3.4 6.8 10.2 13.6
number of bacteria n(t) 1000 1510 2292 3856 5080
(a) find the average rate of change for the number of bacteria from 0 hours to 3.4 hours.
bacteria per hour
(b) find the average rate of change for the number of bacteria from 10.2 hours to 13.6 hours.
bacteria per hour

Explanation:

Step1: Recall average - rate - of - change formula

The average rate of change of a function $y = f(x)$ over the interval $[x_1,x_2]$ is given by $\frac{f(x_2)-f(x_1)}{x_2 - x_1}$. Here, the function is $N(t)$ (number of bacteria) with respect to time $t$ (in hours).

Step2: Solve part (a)

For the interval from $t_1 = 0$ to $t_2=3.4$ hours, $N(0) = 1000$ and $N(3.4)=1510$. The average rate of change is $\frac{N(3.4)-N(0)}{3.4 - 0}=\frac{1510 - 1000}{3.4}=\frac{510}{3.4}=150$ bacteria per hour.

Step3: Solve part (b)

For the interval from $t_1 = 10.2$ to $t_2 = 13.6$ hours, $N(10.2)=3856$ and $N(13.6)=5080$. The average rate of change is $\frac{N(13.6)-N(10.2)}{13.6 - 10.2}=\frac{5080 - 3856}{3.4}=\frac{1224}{3.4}=360$ bacteria per hour.

Answer:

(a) 150 bacteria per hour
(b) 360 bacteria per hour