1. **State the problem:** We are given a function with values at specific points and asked to find on which interval the average rate of change is greatest. 2. **Recall the formula for average rate of change:** $$\text{Average rate of change} = \frac{f(b) - f(a)}{b - a}$$ where $[a,b]$ is the interval. 3. **Calculate the average rate of change for each interval:** - For $[0,3]$: $$\frac{15 - 4}{3 - 0} = \frac{11}{3} \approx 3.67$$ - For $[3,12]$: $$\frac{20 - 15}{12 - 3} = \frac{5}{9} \approx 0.56$$ - For $[12,20]$: $$\frac{27 - 20}{20 - 12} = \frac{7}{8} = 0.875$$ - For $[20,24]$: $$\frac{40 - 27}{24 - 20} = \frac{13}{4} = 3.25$$ 4. **Compare the values:** - $3.67$ for $[0,3]$ - $0.56$ for $[3,12]$ - $0.875$ for $[12,20]$ - $3.25$ for $[20,24]$ 5. **Conclusion:** The greatest average rate of change is on the interval $[0,3]$ with approximately $3.67$. **Final answer:** The average rate of change is greatest on the interval $[0,3]$.