To determine the equations of the racers, we typically need information about their head starts and their speeds. Since I cannot see the graph, I'll guide you through how to write their equations if you had that information.
Let's assume:
- Baby Einstein starts at a certain distance (head start) and has a certain speed.
- Toddler Newton starts from the starting line and has a different speed.
- Pre-teen Darwin also starts at a certain distance (either a head start or from the starting line) and has yet another speed.
You could write their distance equations in the form \(y = mx + b\), where \(m\) is the speed (rate of distance covered) and \(b\) is the head start (initial distance).
Part A: Determine equations for each racer
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Baby Einstein:
- Assume he has a head start of \(b_1\) yards and travels at a speed of \(m_1\) yards/second.
- His equation would be: \[ y = m_1x + b_1 \]
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Toddler Newton:
- Assume he starts from 0 yards and travels at a speed of \(m_2\) yards/second.
- His equation would be: \[ y = m_2x \]
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Pre-teen Darwin:
- Assume he has a head start of \(b_3\) yards and travels at a speed of \(m_3\) yards/second.
- His equation would be: \[ y = m_3x + b_3 \]
Part B: Determine when they are the same distance
To find out when they are the same distance from the finish line, set the equations equal to each other.
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Set Baby Einstein's equation equal to Toddler Newton's: \[ m_1x + b_1 = m_2x \]
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Next, set the equation of Toddler Newton equal to Pre-teen Darwin's: \[ m_2x = m_3x + b_3 \]
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Finally, set Baby Einstein’s equation equal to Pre-teen Darwin's: \[ m_1x + b_1 = m_3x + b_3 \]
By solving these equations, you can find the value of \(x\) (time in seconds) that indicates when they have traveled the same distance. You can substitute \(x\) back into any of the distance equations to find out how far they have traveled.
If you can provide the specific values from the graph, I can assist you more concretely in finding the exact equations and the time when they are equal distances.