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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.





(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=
𝑦
=

Toddler Newton: y=
𝑦
=

Pre-teen Darwin: y=
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=



Part B: Baby Einstein, Toddler Newton, and Pre-teen Darwin will be the same distance

3 answers

To determine the equations for Baby Einstein, Toddler Newton, and Pre-teen Darwin based on the graph provided (which is not visible here), you would typically look for the slope (rate of speed) and the y-intercept (starting position) for each racer. The equations usually take the form \(y = mx + b\), where \(m\) is the slope (speed), and \(b\) is the y-intercept (initial distance).

Part A: Determine equations for each racer

  1. Baby Einstein: If Baby Einstein starts at a distance of \(b\) yards and travels at a constant speed of \(m_E\) yards/second, the equation would be: \[ y = m_E x + b_E \]

  2. Toddler Newton: If Toddler Newton has a different starting position of \(b_T\) yards and he travels at a different speed \(m_T\), the equation would be: \[ y = m_T x + b_T \]

  3. Pre-teen Darwin: If Pre-teen Darwin starts at a distance of \(b_D\) yards and has the fastest speed \(m_D\), the equation would be: \[ y = m_D x + b_D \]

You might need to plug in the specific values from the graph based on the distances and times given.

Part B: Determine when they will be the same distance from the finish line

To find out when all three racers will be the same distance from the finish line, you would set their equations equal to each other and solve for \(x\).

  1. Set \(y_{BE} = y_{TN}\): \[ m_E x + b_E = m_T x + b_T \]

  2. Set \(y_{BE} = y_{PD}\): \[ m_E x + b_E = m_D x + b_D \]

  3. Set \(y_{TN} = y_{PD}\): \[ m_T x + b_T = m_D x + b_D \]

By solving these equations simultaneously, you would find the value(s) of \(x\) (time in seconds) when they will all be equal distances from the finish line.

Please provide the specific values from the graph if you would like help with the coefficients and to perform the actual calculations.

einstein= (8,8)
Newton= (6,8)
darwan=(2,8)

Based on the points provided for each racer, we can derive the equations assuming these points represent positions (in yards) at specific times (in seconds). We can represent the distances traveled as a linear function of time.

Given Points:

  • Baby Einstein: (8, 8) - This implies that at \(x = 0\) seconds, \(y = 8\) yards (an initial distance of 8 yards).
  • Toddler Newton: (6, 8) - This implies that at \(x = 0\) seconds, \(y = 8\) yards (an initial distance of 8 yards).
  • Pre-teen Darwin: (2, 8) - This implies that at \(x = 0\) seconds, \(y = 8\) yards (an initial distance of 8 yards).

Finding the Equations:

From the information, it looks like they all start at the same y-coordinate at time \(x = 0\) (in this case, 8 yards).

Let's assume:

  • Baby Einstein has a speed of certain yards/second.
  • Toddler Newton has a different speed.
  • Pre-teen Darwin has yet another different speed.

To write their equations, we need to deduce their speeds from the provided points. However, it looks like you only provided one point per racer and they all have the same distance at the same time. Typically, we need at least two points for each racer to determine the slope (speed).

Since you only provided a single point each, let’s assume their final positions after a specific time.

For the sake of establishing an example based on typical racing data:

  1. Assume Baby Einstein finishes the race in 8 seconds reaching a distance of 64 yards, so: \[ y = \frac{64 - 8}{8}x + 8 \quad \text{or} \quad y = 7x + 8 \quad \text{(speed of 7 yards/sec)} \]

  2. Assume Toddler Newton finishes the race in 6 seconds reaching a distance of 36 yards, so: \[ y = \frac{36 - 8}{6}x + 8 \quad \text{or} \quad y = \frac{28}{6}x + 8 \quad \text{(speed of about 4.67 yards/sec)} \]

  3. Assume Pre-teen Darwin finishes the race in 2 seconds reaching a distance of 16 yards, so: \[ y = \frac{16 - 8}{2}x + 8 \quad \text{or} \quad y = 4x + 8 \quad \text{(speed of 4 yards/sec)} \]

Resulting Equations:

Based on these assumptions:

  1. Baby Einstein: \(y = 7x + 8\)
  2. Toddler Newton: \(y \approx 4.67x + 8\)
  3. Pre-teen Darwin: \(y = 4x + 8\)

Now you can equate these equations in Part B to find the time at which they are at the same distance from the finish line:

Part B: Setting Equations Equal

To determine when they are at the same distance:

  1. Set Baby Einstein equal to Toddler Newton: \[ 7x + 8 = 4.67x + 8 \] Simplifying: \[ 2.33x = 0 \quad \Rightarrow \quad x = 0 \]

  2. Set Baby Einstein equal to Pre-teen Darwin: \[ 7x + 8 = 4x + 8 \] Simplifying: \[ 3x = 0 \quad \Rightarrow \quad x = 0 \]

  3. Set Toddler Newton equal to Pre-teen Darwin: \[ 4.67x + 8 = 4x + 8 \] Simplifying: \[ 0.67x = 0 \quad \Rightarrow \quad x = 0 \]

Conclusion:

All of the calculations show that the only point where Baby Einstein, Toddler Newton, and Pre-teen Darwin are at the same distance from the finish line (as per the points provided) is at the starting point (0 seconds). For further clarifying details (like speeds and finish positions), more data points would be needed. If there are finish times/distances for each racer, please provide that information to derive exact equations and solve accurately for the distance equivalences!