Question

firing a piece of pottery in a kiln takes place at different temperatures for different amounts of time. during which time interval did the temperature in the kiln show the greatest average rate of change? 1. 0 to 1 hour 2. 1 hour to 1.5 hours 3. 2.5 hours to 5 hours 4. 5 hours to 8 hours

Explanation:

Step1: Recall average - rate - of - change formula

The average rate of change of a function $y = f(x)$ over the interval $[a,b]$ is $\frac{f(b)-f(a)}{b - a}$. Here, the function represents temperature $T$ as a function of time $t$, and the average rate of change of temperature over the time - interval $[t_1,t_2]$ is $\frac{T(t_2)-T(t_1)}{t_2 - t_1}$.

Step2: Calculate rate for 0 to 1 hour

We have the points $(0,0)$ (assume starting temperature is 0) and $(1,700)$. The average rate of change is $\frac{700 - 0}{1-0}=700$ degrees per hour.

Step3: Calculate rate for 1 to 1.5 hours

The points are $(1,700)$ and $(1.5,900)$. The average rate of change is $\frac{900 - 700}{1.5 - 1}=\frac{200}{0.5}=400$ degrees per hour.

Step4: Calculate rate for 2.5 to 5 hours

The points are $(2.5,1300)$ and $(5,1640)$. The average rate of change is $\frac{1640 - 1300}{5 - 2.5}=\frac{340}{2.5}=136$ degrees per hour.

Step5: Calculate rate for 5 to 8 hours

The points are $(5,1640)$ and $(8,1800)$. The average rate of change is $\frac{1800 - 1640}{8 - 5}=\frac{160}{3}\approx53.33$ degrees per hour.

Answer:

1. 0 to 1 hour