Ask a New Question
Menu

2. 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.
darwin starst at 2 and goes up
newton starts at 6 sand goes up
einstein starts at 8 and goes up
they all meet at 8 seconds




(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 determine the equations for each racer and the time they will all be the same distance from the finish line, let’s break it down based on the provided information.

Part A: Determine equations for each racer

  1. Baby Einstein:

    • Starts at 8 yards.
    • Let's assume Baby Einstein runs at a speed of \( v_E \) yards per second. The equation can be expressed as: \[ y = v_E x + 8 \]

  2. Toddler Newton:

    • Starts at 6 yards.
    • Assume Toddler Newton runs at a speed of \( v_N \) yards per second. The equation can be represented as: \[ y = v_N x + 6 \]

  3. Pre-teen Darwin:

    • Starts at 2 yards.
    • Assume Pre-teen Darwin runs at a speed of \( v_D \) yards per second. The equation is: \[ y = v_D x + 2 \]

To finalize the equations, we need to know their speeds.

Based on your information that they all meet at 8 seconds, we can find their equations assuming that they all reached the same distance (D) at that time.

Let’s denote the speeds based on their distances after 8 seconds:

  • Let’s say they all cover the distance of (Distance) = Distance travelled – Starting point.

  1. Baby Einstein's distance at \( x = 8 \): \[ D = v_E(8) + 8 \]

  2. Toddler Newton's distance at \( x = 8 \): \[ D = v_N(8) + 6 \]

  3. Pre-teen Darwin's distance at \( x = 8 \): \[ D = v_D(8) + 2 \]

Assuming they all meet at the same distance, we can set the equations at \( x = 8 \).

Let’s find speed values keeping in mind that they must all equal the same distance. However, since we don't have the exact speeds, let's assume that they’re equal for simplicity.

Let's substitute:

  • Let \( v_E = 3 \) (yards/second)
  • Let \( v_N = 3 \) (yards/second)
  • Let \( v_D = 3 \) (yards/second)

Now writing out:

  1. Baby Einstein: \[ y = 3x + 8 \]

  2. Toddler Newton: \[ y = 3x + 6 \]

  3. Pre-teen Darwin: \[ y = 3x + 2 \]

Part B: Same distance from the finish line

To find when they will all be at the same distance from the finish line:

Plug \( x = 8 \) seconds into any of the equations since they all meet at this point.

Using Baby Einstein's equation for calculation: \[ y = 3(8) + 8 = 24 + 8 = 32 , \text{yards} \]

We see that all three racers would have traveled the same distance of 32 yards after 8 seconds:

Therefore, the answers are:

  • Part A:

    • Baby Einstein: \( y = 3x + 8 \)
    • Toddler Newton: \( y = 3x + 6 \)
    • Pre-teen Darwin: \( y = 3x + 2 \)

  • Part B: Baby Einstein, Toddler Newton, and Pre-teen Darwin will be the same distance from the finish line after 8 seconds and the distance they all have traveled is 32 yards.