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.

Question

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=
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=

Toddler Newton: y=
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=

Pre-teen Darwin: y=
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=



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

Answer 1

To derive the equations for each racer based on the provided details and graph, I will need specific values such as slopes (speeds) and y-intercepts (head starts) from the graph. Since I cannot see the graph, I will guide you on how to find the equations and how to determine when the distances are equal.

Part A: Determine equations for each racer.

The general form of a linear equation in y-intercept form is: \[ y = mx + b \] where:

• \( y \) is the distance from the finish line,
• \( m \) is the slope (speed, in yards per second),
• \( x \) is time (in seconds),
• \( b \) is the y-intercept (head start distance in yards).

If you derive these values from the graph paper, you could write:

1. For Baby Einstein:

   • Let’s say:
     • Head start (b_E): 10 yards
     • Speed (m_E): 3 yards/second
   • Then the equation would be: \[ y = 3x + 10 \]

2. For Toddler Newton:

   • Let’s say:
     • Head start (b_N): 20 yards
     • Speed (m_N): 2 yards/second
   • Then the equation would be: \[ y = 2x + 20 \]

3. For Pre-teen Darwin:

   • Let’s say:
     • Head start (b_D): 5 yards
     • Speed (m_D): 4 yards/second
   • Then the equation would be: \[ y = 4x + 5 \]

Summary of Part A

• Baby Einstein: \( y = 3x + 10 \)
• Toddler Newton: \( y = 2x + 20 \)
• Pre-teen Darwin: \( y = 4x + 5 \)

Part B: Finding when they are the same distance from the finish line.

To find out when all three racers are the same distance from the finish line, we need to set the equations equal to each other:

1. Set Baby Einstein and Toddler Newton equal to each other: \[ 3x + 10 = 2x + 20 \]

   • Solving for \( x \): \[ 3x - 2x = 20 - 10 \] \[ x = 10 \]

2. Now set Baby Einstein equal to Pre-teen Darwin: \[ 3x + 10 = 4x + 5 \]

   • Solving for \( x \): \[ 3x - 4x = 5 - 10 \] \[ -x = -5 \] \[ x = 5 \]

3. Now set Toddler Newton equal to Pre-teen Darwin: \[ 2x + 20 = 4x + 5 \]

   • Solving for \( x \): \[ 2x - 4x = 5 - 20 \] \[ -2x = -15 \] \[ x = 7.5 \]

Next, you must find a common time when all three distances equal by finding a common point of intersection through substitution or graphical methods.

Conclusion

I cannot provide specific distance values as they depend on the equations you derive from the actual graph, so please replace those slopes and intercepts with what you measure from the graph.

Feel free to share those values, and I can help finalize the answer!

Answer 2

As a decimal!

Answer 3

Understood! Let’s directly format all our equations and numerical values into decimal form.

Part A: Determine equations for each racer (in decimal form)

To formulate the equations clearly and ensure they are in decimal form, we can revisit the hypothetical assumptions about the speed and head start. Let’s express everything explicitly in a decimal format.

1. For Baby Einstein:

   • Assume:
     • Speed: 3.0 yards/second
     • Head start: 10.0 yards
   • Equation: \[ y = 3.0x + 10.0 \]

2. For Toddler Newton:

   • Assume:
     • Speed: 2.0 yards/second
     • Head start: 20.0 yards
   • Equation: \[ y = 2.0x + 20.0 \]

3. For Pre-teen Darwin:

   • Assume:
     • Speed: 4.0 yards/second
     • Head start: 5.0 yards
   • Equation: \[ y = 4.0x + 5.0 \]

Summary of Part A in Decimal Form

• Baby Einstein: \( y = 3.0x + 10.0 \)
• Toddler Newton: \( y = 2.0x + 20.0 \)
• Pre-teen Darwin: \( y = 4.0x + 5.0 \)

Part B: Finding when they are the same distance from the finish line.

We set each equation equal to find out after how many seconds they will be at the same distance:

1. Setting Baby Einstein and Toddler Newton equal: \[ 3.0x + 10.0 = 2.0x + 20.0 \] \[ 3.0x - 2.0x = 20.0 - 10.0 \] \[ 1.0x = 10.0 \implies x = 10.0 , \text{seconds} \]

2. Setting Baby Einstein and Pre-teen Darwin equal: \[ 3.0x + 10.0 = 4.0x + 5.0 \] \[ 3.0x - 4.0x = 5.0 - 10.0 \] \[ -1.0x = -5.0 \implies x = 5.0 , \text{seconds} \]

3. Setting Toddler Newton and Pre-teen Darwin equal: \[ 2.0x + 20.0 = 4.0x + 5.0
   \] \[ 2.0x - 4.0x = 5.0 - 20.0
   \] \[ -2.0x = -15.0 \implies x = 7.5 , \text{seconds} \]

Conclusion for Part B

The different pairs meet at the following times:

• Baby Einstein and Toddler Newton meet at 10.0 seconds.
• Baby Einstein and Pre-teen Darwin meet at 5.0 seconds.
• Toddler Newton and Pre-teen Darwin meet at 7.5 seconds.

Thus, at distinct times, each pair reaches the same distance from the finish line, but they do not meet altogether at the same distance simultaneously based on these example values.

If you'd like to recalculate with different numbers or use actual graph measurements, please share those details!