Math Problem Statement

The speed, in miles per hour, of a race car on a track is modeled by the linear function v(t)=60+1.2t, where t is the number of seconds since the car began its second lap around the track.

Solution

The function v(t)=60+1.2tv(t) = 60 + 1.2t models the speed of a race car in miles per hour, where tt is the time in seconds since the car started its second lap.

Key Points:

  1. Initial Speed: At t=0t = 0, the speed is: v(0)=60+1.2(0)=60 mphv(0) = 60 + 1.2(0) = 60 \text{ mph}

  2. Rate of Change: The term 1.2t1.2t indicates that the speed increases by 1.2 miles per hour for each second that passes.

  3. Speed After Time: To find the speed at any time tt, simply plug in the value of tt into the function. For example:

    • At t=10t = 10 seconds: v(10)=60+1.2(10)=60+12=72 mphv(10) = 60 + 1.2(10) = 60 + 12 = 72 \text{ mph}
    • At t=30t = 30 seconds: v(30)=60+1.2(30)=60+36=96 mphv(30) = 60 + 1.2(30) = 60 + 36 = 96 \text{ mph}
  4. Maximizing Speed: The function is linear, meaning there is no maximum speed as tt increases; the speed continues to increase indefinitely.

Questions for Further Exploration:

  1. What is the speed of the car after 5 minutes?
  2. How long will it take for the car to reach a speed of 100 mph?
  3. What would the speed be if the car maintained this rate for 1 hour?
  4. How does this model compare to a constant speed model?
  5. Can you describe how this model could be used to predict lap times?

Tip:

Always remember that in linear models, the slope indicates the rate of change, and the y-intercept represents the initial value.

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Math Problem Analysis

Mathematical Concepts

Linear Functions
Rate of Change
Speed and Time

Formulas

v(t) = 60 + 1.2t

Theorems

Linear Function Properties

Suitable Grade Level

Grades 8-10