Question

graph the absolute value equation that represents the given situation, $d = \frac{1}{5}|s - 250| - 50$. then mark the points that represent the horizontal distance from the left shore where the river bottom is 20 feet below the surface.

Explanation:

Step1: Set d equal to -20

$d = -20$, so substitute into the equation:
$$-20 = \frac{1}{8}|s - 250| - 50$$

Step2: Isolate the absolute value term

Add 50 to both sides:
$$-20 + 50 = \frac{1}{8}|s - 250|$$
$$30 = \frac{1}{8}|s - 250|$$

Step3: Eliminate the fraction

Multiply both sides by 8:
$$30 \times 8 = |s - 250|$$
$$240 = |s - 250|$$

Step4: Solve absolute value equation

Split into two cases:
Case 1: $s - 250 = 240$
$$s = 240 + 250 = 490$$
Case 2: $s - 250 = -240$
$$s = -240 + 250 = 10$$

Step5: Identify vertex of the graph

For $d = \frac{1}{8}|s - 250| - 50$, the vertex is at $s=250$, $d=-50$, so the point is $(250, -50)$. To graph the V-shaped absolute value function, plot the vertex, then use the slope $\frac{1}{8}$ and $-\frac{1}{8}$ for the two rays from the vertex.

Answer:

The points where the river bottom is 20 feet below the surface are $(10, -20)$ and $(490, -20)$. The absolute value graph has a vertex at $(250, -50)$, with one ray going from $(250, -50)$ with slope $\frac{1}{8}$ (e.g., to $(330, -40)$) and the other ray going from $(250, -50)$ with slope $-\frac{1}{8}$ (e.g., to $(170, -40)$).