Problem 62 - Algebra 1

Question

In Exercises 62 and \(63,\) a mountain climber is scaling a 300 -foot cliff at a constant rate. The climber starts at the bottom at 12: 00 P.M. By 12: 30 P.M., the climber has moved 62 feet up the cliff. Write an equation that gives the distance \(d\) (in feet) remaining in the climb in terms of the time \(t\) (in hours), What is the slope of the line?

Step-by-Step Solution

Answer

The equation that gives the distance \(d\) (in feet) remaining in the climb in terms of the time \(t\) (in hours) is \(d = 300 - 124t\), and the slope of the line is -124.

Step 1: Determine the Climber's Speed

First, we'll figure out the velocity of the climber, which can be determined from the data provided. In 0.5 hours (12:30 PM -12:00 PM), the climber scales 62 feet. So, the speed will be \(speed = \frac{distance}{time} = \frac{62}{0.5} = 124\) feet/hour.

Step 2: Formulate the Equation

The equation must reflect that the remaining distance decreases as the climber moves up. At t=0, the distance to be covered is 300 feet. After that, every hour the climber scales 124 feet, or in terms of t, it scales \(124t\) feet. Since the equation reflects the remaining distance, we subtract the distance covered so far from the total distance of 300 feet. Therefore, the equation becomes \(d = 300 - 124t\).

Step 3: Find the Slope

The slope (m) of the line is the coefficient of \(t\) in the equation. In \(d = 300 - 124t\), the slope is -124.

Key Concepts

Distance-Time Relationship

The distance-time relationship describes how the remaining distance a mountain climber has left to scale changes over time. This is crucial because it allows us to predict the climber's position at any given moment. In this scenario, we have a 300-foot cliff, and a climber starting at the base at 12:00 PM. By 12:30 PM, the climber has ascended 62 feet.

• To understand this relationship, we represent the total distance to be covered as 300 feet.
• The remaining distance decreases as the climber ascends the cliff.
• We need to express the remaining distance at any time \( t \), where \( t \) is measured in hours from 12:00 PM.

Thus, the distance remaining is dynamically linked to the time that has passed since the climber started, following the pattern seen in this relationship.

Rate of Change

In our problem, the rate of change refers to the speed at which the climber ascends the cliff. Here, it is essential to calculate how fast the remaining distance diminishes over time. The rate of change is represented by how many feet the climber scales in an hour, also known as the velocity.

• From the data, in half an hour, the climber covers 62 feet.
• Calculating the velocity involves determining feet per hour, resulting in \( \frac{62}{0.5} = 124 \) feet per hour.
• This rate of change is crucial for predicting how long it will take to finish the climb and for formulating the linear equation.

The rate of change concept helps in understanding how quickly the remaining distance reduces as time progresses, providing a clear picture of the climber's pace.

Equation Formulation

Formulating the equation is the process of creating a linear equation that accurately describes the relationship between distance and time for this specific scenario. Let's break down the steps:
Starting with the initial total distance of 300 feet, we need to consider how the climber's ascent reduces this distance over time. Since the climber ascends at a rate of 124 feet per hour, the equation must account for this ascent by subtracting \( 124t \), which represents the distance covered in \( t \) hours.

• The initial distance is 300 feet, which decreases as the climber ascends.
• To represent the remaining distance \( d \), we use the formula: \( d = 300 - 124t \).
• This equation shows the effect of time on the remaining distance, with the slope of the equation, \(-124\), indicating the climber's rate of ascent.

Thus, equation formulation is about translating the physical activity of climbing into a mathematical representation that captures how distance decreases over time.