Question

1. the depth of a river changes after a heavy rainstorm. its depth, in feet, is modeled as a function of time, in hours. consider this graph of the function. estimate, to the nearest whole number, the average rate of change (slope) of the depth of the river for the first 18 hours. the estimated average rate of change for the first 18 hours is ft/hour.

Explanation:

Step1: Find the initial and final values

The formula for the average rate of change (slope) of a function \(y = f(x)\) over the interval \([x_1,x_2]\) is \(\frac{f(x_2)-f(x_1)}{x_2 - x_1}\).
For the first 18 hours, \(x_1=0\) (initial time) and \(x_2 = 18\) (final time).
From the graph, when \(x = 0\), \(y=f(0)=9\) (initial depth) and when \(x = 18\), \(y=f(18)=21\) (final depth).

Step2: Calculate the average rate of change

Substitute the values into the formula: \(\frac{f(18)-f(0)}{18 - 0}=\frac{21 - 9}{18}\).
Simplify the numerator: \(21-9 = 12\).
So the expression becomes \(\frac{12}{18}=\frac{2}{3}\approx0.67\). But wait, no! Wait, looking at the graph again, if we assume the grid - based on the \(y\) - axis (depth) and \(x\) - axis (time).
Wait, another approach:
The average rate of change formula is \(\text{Average Rate of Change}=\frac{\Delta y}{\Delta x}\).
\(\Delta y\) (change in depth) and \(\Delta x\) (change in time).
From \((0,9)\) to \((18,21)\)
\(\Delta y=21 - 9=12\), \(\Delta x=18-0 = 18\)
\(\text{Average Rate of Change}=\frac{12}{18}=\frac{2}{3}\approx0.67\). But wait, no! Wait, if we consider the grid:
Looking at the \(y\) - axis (depth) and \(x\) - axis (time).
If we assume that the points are \((0,9)\) and \((18,20)\) (by visual estimation of the graph, since \(21\) is a bit high, maybe a mis - read. If we take \(y\) at \(x = 18\) as \(20\))
\(\text{Average Rate of Change}=\frac{20 - 9}{18}=\frac{11}{18}\approx0.61\). But another way:
The formula for average rate of change of a function \(y=f(x)\) over \([a,b]\) is \(\frac{f(b)-f(a)}{b - a}\)
Here \(a = 0\), \(b=18\)
From the graph, \(f(0)=9\) (exact), and \(f(18)\approx20\) (by visual inspection of the graph, since the point at \(x = 18\) is just below \(21\))
\(\frac{20 - 9}{18}=\frac{11}{18}\approx0.61\). But if we use the formula correctly with the graph - reading:
The two points \((0,9)\) and \((18,20)\) (estimated from the graph)
\(\text{Average Rate of Change}=\frac{20 - 9}{18}=\frac{11}{18}\approx0.61\). But wait, no! Wait, if we consider the grid: each square on the \(y\) - axis (depth) is \(3\) feet (\(24-21 = 3\), \(21 - 18=3\) etc.) and on the \(x\) - axis each square is \(3\) hours (\(3-0 = 3\), \(6 - 3=3\) etc.)
If at \(x = 18\), \(y\) is \(20\) (since \(18\) on \(x\) - axis: \(18\div3 = 6\) squares from \(x = 0\); and \(y\) at \(x = 18\) is \(20\) (approx \(6 + \frac{2}{3}\) squares of \(3\) feet: \(6\times3=18\), \(\frac{2}{3}\times3 = 2\), total \(20\))
\(\text{Average Rate of Change}=\frac{20 - 9}{18}=\frac{11}{18}\approx0.61\approx1\) (to the nearest whole number)

Answer:

\(1\)