Question

learning check #1
a sprinkler is spraying a stream of water into the air. the solid portion of the graph represents the path of the water, where x represents the distance in feet from the sprinkler and y represents the height in feet of the water.
find the rate of change over the interval 0 ≤ x ≤ 4 and interpret the result.

Explanation:

Step1: Identify the formula for average rate of change

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, $a = 0$, $b = 4$.

Step2: Determine the values of $y$ at $x = 0$ and $x=4$ from the graph

From the graph, when $x = 0$, $y=0$ (the water starts at the sprinkler's height which we can assume is ground - level, so $y(0)=0$). When $x = 4$, $y = 6$.

Step3: Calculate the rate of change

Using the formula $\frac{y(4)-y(0)}{4 - 0}=\frac{6 - 0}{4}=\frac{6}{4}=1.5$.

Step4: Interpret the result

The rate of change of $1.5$ means that, on average, for every 1 - foot increase in the horizontal distance $x$ from the sprinkler in the interval $0\leq x\leq4$, the height $y$ of the water increases by 1.5 feet.

Answer:

The rate of change over the interval $0\leq x\leq4$ is $1.5$ feet per foot. This means that, on average, the height of the water increases by 1.5 feet for every 1 - foot increase in the horizontal distance from the sprinkler in the given interval.