Question

a group of climbers begin climbing at an elevation of 7000 feet and ascend at a steady rate of 500 vertical feet per hour. this situation can be modeled by a linear function. identify the independent and dependent variables. draw a graph of the function, then use the graph to find their elevation after 4 hours. is a linear model reasonable for this situation? the number of hours is the variable. the elevation in feet is the variable.

Explanation:

Step1: Identify Variables

In a relationship, the independent variable is the one that is manipulated or changes freely (here, time in hours, as it progresses on its own). The dependent variable depends on the independent one (elevation depends on hours climbed). So hours: independent; elevation: dependent.

Step2: Linear Function Setup

The linear function for elevation \( E \) (in feet) over time \( t \) (in hours) is \( E = 7000 + 500t \).

Step3: Elevation at 4 Hours

Substitute \( t = 4 \) into the function: \( E = 7000 + 500(4) = 7000 + 2000 = 9000 \) feet.

Step4: Reasonableness of Linear Model

A linear model is reasonable here because the climbers ascend at a steady rate (constant slope, no acceleration/deceleration or changes in rate mentioned), so elevation increases linearly with time.

Step5: Graphing (Brief)

• x - axis: Time (hours), from 0 to, say, 5.
• y - axis: Elevation (feet), from 7000 upwards.
• Plot the y - intercept at \( (0, 7000) \), then use the slope (500) to find other points (e.g., \( (1, 7500) \), \( (2, 8000) \), etc.), and draw a straight line. From the graph, at \( t = 4 \), the y - value (elevation) is 9000, matching the calculation.

Answer:

• Independent variable: number of hours (independent)
• Dependent variable: elevation in feet (dependent)
• Elevation after 4 hours: 9000 feet
• Linear model: Reasonable (since the ascent rate is steady, elevation changes linearly with time)