Question

use the scenario to graph the equation and answer the questions.

1. at hickory hollow ranch, a drilling team detected water 38 feet below the ground. if it took workers 8 days to be able to dig and pull up the water, how much did they dig each day on average? draw a graph to represent the situation. what do the x- and y- intercepts represent?

Explanation:

Step1: Define Variables

Let \( x \) be the number of days, and \( y \) be the depth (in feet) below the ground. The total depth is 38 feet, and it takes 8 days. The rate (depth per day) is constant, so the equation is linear. The slope \( m \) (average depth dug per day) can be found by \( m=\frac{\text{change in }y}{\text{change in }x} \). We start at \( y = - 38 \) (38 feet below ground, so negative) when \( x = 0 \), and end at \( y = 0 \) (ground level) when \( x = 8 \). So the slope \( m=\frac{0 - (-38)}{8 - 0}=\frac{38}{8}=4.75 \). Wait, actually, if we consider the depth being dug, the equation can be \( y=-38 + 4.75x \), where \( y \) is the depth remaining. But to find the average per day, we use total depth divided by total days: \( \text{Average per day}=\frac{38}{8}=4.75 \) feet per day.

Step2: Interpret Intercepts

• y - intercept: When \( x = 0 \) (day 0), \( y=-38 \). This represents the initial depth of the water below the ground (38 feet below ground).
• x - intercept: When \( y = 0 \) (depth is 0, at ground level), we solve \( 0=-38 + 4.75x \), \( 4.75x = 38 \), \( x = 8 \). This represents the number of days it takes to reach the water (dig to ground level from the water's depth).

Answer:

The workers dug an average of \( 4.75 \) feet per day. The \( y \)-intercept (\( (0, - 38) \)) represents the initial depth of the water (38 feet below ground), and the \( x \)-intercept (\( (8, 0) \)) represents the number of days (8 days) it takes to dig to the water (reach ground level from the water's depth). For the graph, plot the points \( (0, - 38) \) and \( (8, 0) \) and draw a line through them.