Question

the average rate of change from x = 0 to x = 15 is about -4.667. how does the average rate of change from x = 0 to x = 20 compare to this number? the average rate of change from x = 0 to x = 20 is decreasing the average rate of change from x = 0 to x = 15.

Explanation:

Step1: Calculate the average rate of change from \(x = 0\) to \(x=20\)

The formula for 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, \(a = 0\), \(f(0)=80\), \(b = 20\), \(f(20)=10\).
So, the average rate of change is \(\frac{10 - 80}{20-0}=\frac{-70}{20}=-3.5\)

Step2: Compare the two average rate of change values

We have two average - rate - of - change values: \(-4.667\) (from \(x = 0\) to \(x = 15\)) and \(-3.5\) (from \(x = 0\) to \(x = 20\)).
Since \(-3.5> - 4.667\) (because when comparing two negative numbers, the number with a smaller magnitude is larger. For example, \(-3.5=-\frac{7}{2}\) and \(-4.667 =-\frac{14}{3}\), and \(\frac{7}{2}=\frac{21}{6}\), \(\frac{14}{3}=\frac{28}{6}\), \(\frac{21}{6}<\frac{28}{6}\) so \(-\frac{21}{6}>-\frac{28}{6}\))

Answer:

The average rate of change from \(x = 0\) to \(x = 20\) (\(-3.5\)) is greater than the average rate of change from \(x = 0\) to \(x = 15\) (\(-4.667\))