To provide the requested equations for each racer, I would typically analyze a graph representing their distances over time. However, since I don't have access to that graph, I will guide you through a hypothetical scenario to derive the equations based on common principles.
Part A: Determine equations for each racer
To derive the equations for each racer, we assume:
- Each racer's distance over time can be modeled by a linear equation in the form of \( y = mx + b \), where:
- \( y \) represents the distance from the finish line (in yards).
- \( x \) represents the time (in seconds).
- \( m \) represents the slope (rate of speed).
- \( b \) is the y-intercept (the head start each racer has).
Let's assume hypothetical values:
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Baby Einstein starts with a head start of 5 yards and runs at a speed of 3 yards per second. \[ y = 3x + 5 \]
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Toddler Newton starts with a head start of 10 yards and runs at a speed of 2 yards per second. \[ y = 2x + 10 \]
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Pre-teen Darwin starts with no head start and runs at a speed of 4 yards per second. \[ y = 4x + 0 \quad \text{or simply} \quad y = 4x \]
Part B: Finding when they are the same distance
To find when all three are the same distance from the finish line, we set their equations equal to each other.
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Set Baby Einstein equal to Toddler Newton: \[ 3x + 5 = 2x + 10 \] \[ 3x - 2x = 10 - 5 \] \[ x = 5 \text{ seconds} \]
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Set Toddler Newton equal to Pre-teen Darwin: \[ 2x + 10 = 4x \] \[ 10 = 4x - 2x \] \[ 2x = 10 \] \[ x = 5 \text{ seconds} \]
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Set Baby Einstein equal to Pre-teen Darwin: \[ 3x + 5 = 4x \] \[ 5 = 4x - 3x \] \[ x = 5 \text{ seconds} \]
All the equations confirm that after 5 seconds, all three racers are the same distance from the finish line.
Calculate Distance Traveled
To determine the distance traveled after 5 seconds:
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For Baby Einstein: \( y = 3(5) + 5 = 15 + 5 = 20 \text{ yards} \)
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For Toddler Newton: \( y = 2(5) + 10 = 10 + 10 = 20 \text{ yards} \)
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For Pre-teen Darwin: \( y = 4(5) = 20 \text{ yards} \)
Final Answers:
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Part A:
- Baby Einstein: \( y = 3x + 5 \)
- Toddler Newton: \( y = 2x + 10 \)
- Pre-teen Darwin: \( y = 4x \)
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Part B: Baby Einstein, Toddler Newton, and Pre-teen Darwin will be the same distance from the finish line after 5 seconds, and the distance they all have traveled is 20 yards.
Feel free to adjust the slopes and intercepts as necessary with the actual graph data!
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