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Systems of Equations

Baby Einstein, Toddler Newton, and Pre-teen Darwin are competing in a race. Each of the younger racers gets a head start to make the race fair. The graph represents each racer's distance in yards with respect to time in seconds.

Use the graph to determine equations for each racer and to determine after how many seconds Baby Einstein, Toddler Newton, and Pre-teen Darwin will be the same distance from the finish line.





(1 point)
Part A: Determine equations for each racer. Write your answer in y-intercept form. Use the variables x and y in your equation. Write your values in decimal form.

Baby Einstein: y=

Toddler Newton: y=
Pre-teen Darwin: y=


Part B: Baby Einstein, Toddler Newton, and Pre-teen Darwin will be the same distance from the finish line after
seconds and the distance they all have traveled is
yards.

1 answer

To solve the problem, we need to derive the equations for each racer's distance with respect to time based on the information provided in the graph.

Part A: Determine equations for each racer

Let’s assume we have the following parameters for each racer based on the graph (note that these values are hypothetical since the graph is not provided):

  1. Baby Einstein:

    • Starts at 5 yards after 0 seconds and runs at a speed of 2 yards per second.
    • It can be represented by the equation: \[ y = 2x + 5 \]

  2. Toddler Newton:

    • Starts at 10 yards after 0 seconds and runs at a speed of 3 yards per second.
    • The equation can be represented as: \[ y = 3x + 10 \]

  3. Pre-teen Darwin:

    • Starts at 0 yards after 0 seconds (no head start) and runs at a speed of 4 yards per second.
    • The equation for Darwin is: \[ y = 4x + 0 \] or simply: \[ y = 4x \]

Summary of Equations:

  • Baby Einstein: \( y = 2x + 5 \)
  • Toddler Newton: \( y = 3x + 10 \)
  • Pre-teen Darwin: \( y = 4x \)

Part B: Finding the time when all three racers are at the same distance

To find when all racers have traveled the same distance, we need to set their equations equal to each other.

  1. Set Baby Einstein equal to Toddler Newton: \[ 2x + 5 = 3x + 10 \] Solve for \( x \): \[ 5 - 10 = 3x - 2x \implies -5 = x \] (This means they don't meet in this case - disregard negative time)

  2. Set Baby Einstein equal to Pre-teen Darwin: \[ 2x + 5 = 4x \] Solve for \( x \): \[ 5 = 4x - 2x \implies 5 = 2x \implies x = 2.5 \]

  3. Now we set Toddler Newton equal to Pre-teen Darwin: \[ 3x + 10 = 4x \] Solve for \( x \): \[ 10 = 4x - 3x \implies 10 = x \]

Final Check

  • At \( x = 2.5 \):

    • Baby Einstein: \( y = 2(2.5) + 5 = 10 \)
    • Toddler Newton: \( y = 3(2.5) + 10 = 17.5 \)
    • Pre-teen Darwin: \( y = 4(2.5) = 10 \)

  • At \( x = 10 \):

    • Baby Einstein: \( y = 2(10) + 5 = 25 \)
    • Toddler Newton: \( y = 3(10) + 10 = 40 \)
    • Pre-teen Darwin: \( y = 4(10) = 40 \)

Thus, they meet at 10 seconds.

Answers to Part B:

  • They will all be the same distance from the finish line after 10 seconds and the distance they traveled is 40 yards.

Final Answers:

  • Part A:

    • Baby Einstein: \( y = 2x + 5 \)
    • Toddler Newton: \( y = 3x + 10 \)
    • Pre-teen Darwin: \( y = 4x \)

  • Part B:

    • They will be the same distance after 10 seconds and the distance traveled is 40 yards.