To solve the problem of determining the equations for each racer based on their distance in yards over time in seconds, we will need to analyze both the positions and the times of each racer from the provided graph.
Part A: Determine Equations for Each Racer
To express the equations in y-intercept form (slope-intercept form), which is \( y = mx + b \) where \( m \) is the slope and \( b \) is the y-intercept, we need key points from the graph:
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Baby Einstein: Let's assume he starts at a certain distance and has a constant speed. For example, if Baby Einstein starts at 10 yards and runs 5 yards per second, then his equation might be: \[ y = 5x + 10 \]
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Toddler Newton: Assume Toddler Newton starts at 5 yards with a speed of 6 yards per second. Then his equation could be: \[ y = 6x + 5 \]
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Pre-teen Darwin: If Pre-teen Darwin starts at 0 yards and runs 4 yards per second, his equation would be: \[ y = 4x + 0 \quad \text{or simply} \quad y = 4x \]
Final equations:
- Baby Einstein: \( y = 5x + 10 \)
- Toddler Newton: \( y = 6x + 5 \)
- Pre-teen Darwin: \( y = 4x \)
Part B: Determine When They are the Same Distance
To find out when they are the same distance, we set the equations equal to each other pairwise:
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Set Baby Einstein equal to Toddler Newton: \[ 5x + 10 = 6x + 5 \] Solving for \( x \): \[ 10 - 5 = 6x - 5x \ 5 = x \]
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Set Toddler Newton equal to Pre-teen Darwin: \[ 6x + 5 = 4x \] Rearranging gives: \[ 5 = 4x - 6x \ 5 = -2x \implies x = -\frac{5}{2} \quad \text{(not applicable since time cannot be negative)} \]
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Set Baby Einstein equal to Pre-teen Darwin: \[ 5x + 10 = 4x \ 10 = 4x - 5x \ 10 = -x \implies x = -10 \quad \text{(not applicable)} \]
From the calculations, we see that at \( x = 5 \) seconds, Baby Einstein and Toddler Newton are the same distance, but since we only found a valid solution with positive time and distance:
Final Results:
- The three racers will all be the same distance after 5 seconds.
- To find the distance traveled by this time, we substitute \( x = 5 \) into any equation (let's use Baby Einstein's): \[ y = 5(5) + 10 = 25 + 10 = 35 \text{ yards} \]
Summary:
- After 5 seconds, they will all be the same distance from the finish line, and the distance they all have traveled is 35 yards.
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