Question

the graph shows the temperature in a kiln while firing a piece of pottery. over which time interval did the temperature in the kiln have the smallest rate of change?
cannot be determined from the graph
0 to 1.4 hours
2.5 to 4.6 hours
1.4 to 2.5 hours
4.6 to 8.3 hours

Explanation:

Step1: Recall 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, temperature is a function of time, and we calculate the rate of change of temperature with respect to time over each interval.

Step2: Calculate rate of change for 0 to 1.4 hours

Let $(a_1,b_1)=(0,1.4)$ and $(T_1,T_2)=(245,1125)$. Rate of change $r_1=\frac{1125 - 245}{1.4-0}=\frac{880}{1.4}\approx628.57$.

Step3: Calculate rate of change for 1.4 to 2.5 hours

Let $(a_2,b_2)=(1.4,2.5)$ and $(T_3,T_4)=(1125,1300)$. Rate of change $r_2=\frac{1300 - 1125}{2.5 - 1.4}=\frac{175}{1.1}\approx159.09$.

Step4: Calculate rate of change for 2.5 to 4.6 hours

Let $(a_3,b_3)=(2.5,4.6)$ and $(T_5,T_6)=(1300,1675)$. Rate of change $r_3=\frac{1675 - 1300}{4.6 - 2.5}=\frac{375}{2.1}\approx178.57$.

Step5: Calculate rate of change for 4.6 to 8.3 hours

Let $(a_4,b_4)=(4.6,8.3)$ and $(T_7,T_8)=(1675,1925)$. Rate of change $r_4=\frac{1925 - 1675}{8.3 - 4.6}=\frac{250}{3.7}\approx67.57$.

Answer:

4.6 to 8.3 hours