Problem 55 - Calculus with Concepts in Calculus

Question

Suppose you walk along a straight path from your dormitory to the mathematics building, which is 300 meters away. After 100 meters you pass the Student Union. Assuming that you walk at the rate of 1 meter per second, sketch the graph of your distance from the Student Union as a function of time \(t\), with \(t=0\) corresponding to the instant you leave your dormitory.

Step-by-Step Solution

Answer

A V-shaped graph, decreasing then increasing, represents your distance from the Student Union.

Step 1: Understanding the Problem

You're walking along a straight line from your dormitory to the mathematics building, which is 300 meters away, passing the Student Union at 100 meters. You're walking at 1 meter per second, and we need to graph your distance from the Student Union over time, starting from when you leave your dormitory (\(t=0\)).

Step 2: Identify Important Points

At \(t=0\), you start at the dormitory, 100 meters away from the Student Union. After 100 seconds, you reach the Student Union (distance is 0), and after 300 seconds, you reach the mathematics building, 200 meters away from the Student Union.

Step 3: Set Up the Axes

The horizontal axis (x-axis) represents time in seconds. The vertical axis (y-axis) represents the distance from the Student Union in meters.

Step 4: Plot Key Points

Start (\(t=0\), distance 100m), Student Union (\(t=100\), distance 0m), and Mathematics Building (\(t=300\), distance 200m).

Step 5: Sketch the Graph

Draw a straight line from (0, 100) to (100, 0), representing the 100 meters walked towards the Student Union, then a straight line from (100, 0) to (300, 200), representing the 200 meters walked past the Student Union to the mathematics building.

Key Concepts

Straight Line Path

Walking along a straight line is the simplest path one can take. In this exercise, you are walking in a straight line from the dormitory to the mathematics building. This path can be imagined as a direct and uninterrupted line with no turns or detours.

• Your walk starts at the dormitory and ends at the mathematics building.
• The Student Union is a key point along this path, located 100 meters from the dormitory.
• The total length of the path is 300 meters.

This straight line path helps visualize the journey and create a straightforward map for graphing purposes. The path dictates that as you move along it at a constant pace (in this case, 1 meter per second), your changes in position will be linear, leading to straight lines on the graph.

Distance Function

A distance function describes how far something has moved over a certain period relative to a reference point. In this case, the Student Union is our reference point.

• Initially, when you start walking from your dormitory, the Student Union is 100 meters away.
• As you walk towards it, the distance decreases linearly until reaching zero when you arrive at the Student Union after 100 seconds.
• Continuing past the Student Union, the distance starts increasing again, reaching a maximum of 200 meters when you arrive at the mathematics building after 300 seconds.

Thus, this distance function can be represented as two straight-line segments on a graph. The first segment goes downhill from 100 meters to 0 meters, while the second segment ascends from 0 meters to 200 meters. This reflects your steady pace and changing position relative to the Student Union.

Time Representation

Time representation on a graph often involves the horizontal axis (x-axis) displaying intervals of time, while the vertical axis (y-axis) shows the changing variable—in this case, distance from the Student Union.

• The journey starts at time zero ( \( \text{t}=0 \) ) when you leave the dormitory.
• At 100 seconds ( \( \text{t}=100 \) ), you've reached the Student Union.
• By 300 seconds ( \( \text{t}=300 \) ), you arrive at the mathematics building.

Time moves from left to right, mapping the steady progress of your walk. This gives us a visual representation of how distance changes over time.
Each plotted point (such as those at 0, 100, and 300 seconds) denotes a significant moment in the walk, helping us to clearly sketch the graph's lines and slopes efficiently. This graph provides an intuitive snapshot of your journey.