Question

ryan is cooking a roast. the table below gives the temperature ( r(t) ) of the roast (in degrees celsius), at a few times ( t ) (in minutes) after he removed it from the oven.

(a) find the average rate of change for the temperature from 0 minutes to 15 minutes.
( square )°c per minute
(b) find the average rate of change for the temperature from 60 minutes to 75 minutes.
( -3.4 )°c per minute

Explanation:

Step1: Recall the formula for average rate of change

The formula for the average rate of change of a function \(y = f(x)\) over the interval \([x_1,x_2]\) is \(\frac{f(x_2)-f(x_1)}{x_2 - x_1}\).

Step2: Solve part (a)

For part (a), \(x_1 = 0\), \(f(x_1)=193.5\), \(x_2 = 15\), \(f(x_2)=166.5\).
Substitute into the formula: \(\frac{166.5 - 193.5}{15-0}=\frac{- 27}{15}=-1.8\)

Step3: Solve part (b)

For part (b), \(x_1 = 60\), \(f(x_1)=73.5\), \(x_2 = 75\), \(f(x_2)=22.5\).
Substitute into the formula: \(\frac{22.5 - 73.5}{75 - 60}=\frac{-51}{15}=-3.4\)

Answer:

(a) \(-1.8\)°C per minute; (b) \(-3.4\)°C per minute