Math Problem Statement

A cross-country runner is training for a marathon. His training program requires him to run at different speeds for different lengths of time. His program also requires him to accelerate and decelerate at a constant rate. Today he begins by jogging for 10 min at a rate of 5 miles per hour. He then spends 1 min accelerating to a rate of 10 miles per hour. He stays at this rate for 5 min. He then decelerates for 1 min to a rate of 7 miles per hour. He stays at this rate for 30 min. Finally, to cool down, he decelerates for 2 min to a rate of 3 miles per hour. He stays at this rate for a final 10 min and then stops.

Solution

The image contains a problem related to the speed-time behavior of a cross-country runner training for a marathon, with parts (a) through (d) posing various mathematical questions. Here's a breakdown of the problem:

Problem:

A cross-country runner's training involves varying speeds and periods of acceleration and deceleration. His running schedule for today is:

  1. Jogging for 10 minutes at 5 miles per hour.
  2. Accelerating for 1 minute to reach 10 miles per hour.
  3. Running at 10 miles per hour for 5 minutes.
  4. Decelerating for 1 minute to reach 7 miles per hour.
  5. Running at 7 miles per hour for 30 minutes.
  6. Decelerating for 2 minutes to 3 miles per hour.
  7. Running at 3 miles per hour for 10 minutes before stopping.

Tasks:

  • (a) Make a speed vs. time graph representing this situation.
  • (b) Find the instantaneous rate of change in the runner’s speed at 10.5 minutes.
  • (c) Calculate the average rate at which he changed speeds from minute 11 to minute 49.
  • (d) Explain why the answer for part (c) doesn’t fully represent the speed change in the runner's training between minute 11 and minute 49.

Would you like a detailed solution for each part or just specific parts of the problem?

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Rate of Change
Speed-Time Graph
Instantaneous Rate of Change
Average Rate of Change

Formulas

Rate of change = (change in speed) / (change in time)
Instantaneous rate of change using derivative (if applicable)

Theorems

Average Rate Theorem
Concept of Instantaneous Rate of Change in Calculus

Suitable Grade Level

High School (Grades 10-12)

Related Recommendation

Distance-Time Analysis and Rate of Change for Runners

Proving a Runner's Speed at Exactly 10 mph in a 6.2-Mile Race Using Calculus

Determine Fastest Interval in a Distance-Time Graph for a 5-mile Training Run

Analyzing Runners' Velocities from Position-Time Graphs in a 100-Meter Race

Runner's Distance and Speed Using Calculus: d(t)=2/3t^3 - 35/6t^2 + 35/2t