Math Problem Statement

A car moving at a constant speed passed a timing device at t=0. After 6 seconds, the car has traveled 486 ft. Write a linear function rule to model the distance in feet d the car has traveled any number of seconds t after passing the timing device.

Solution

The problem describes a car traveling at a constant speed, passing a timing device at t=0t = 0, and then traveling 486 feet in 6 seconds. To model this situation as a linear function:

Step 1: Recall the structure of a linear equation.

The linear function takes the form: d=mt+bd = mt + b Where:

  • dd: distance (in feet)
  • tt: time (in seconds)
  • mm: the rate of change (speed) in feet per second
  • bb: the y-intercept (distance at t=0t = 0).

Step 2: Use the given information.

  • At t=0t = 0: The car is at the timing device, so b=0b = 0.
  • At t=6t = 6: The car has traveled d=486d = 486 feet.

Step 3: Calculate the speed (mm).

Since the car travels a constant speed: m=Change in distanceChange in time=4866=81ft/sm = \frac{\text{Change in distance}}{\text{Change in time}} = \frac{486}{6} = 81 \, \text{ft/s}

Step 4: Write the linear function.

Substituting m=81m = 81 and b=0b = 0, the equation becomes: d=81td = 81t

Final Answer:

The linear function rule is d=81t.\text{The linear function rule is } d = 81t.

Would you like a deeper explanation or examples of applying this equation?

Related Questions:

  1. How can we interpret the slope m=81m = 81 in this context?
  2. What is the distance traveled after 10 seconds according to the function d=81td = 81t?
  3. How would the function change if the car started 100 feet ahead of the timing device?
  4. If a second car travels at a constant speed of 90 ft/s, how long would it take for it to catch up to the first car?
  5. How would you model this if the car decelerates instead of traveling at a constant speed?

Tip:

Always identify the slope (mm) as the "rate of change" to interpret real-world situations effectively in linear models.

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Math Problem Analysis

Mathematical Concepts

Linear Functions
Rates of Change
Algebra

Formulas

d = mt + b
m = (change in distance) / (change in time)

Theorems

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Suitable Grade Level

Grades 7-9