To write the equations in y-intercept form (y = mx + b), we need the speed of each racer (slope, m) and their starting distance from the finish line (y-intercept, b). Since you did not provide specific speeds or starting distances, I'll create an example using hypothetical values.
Part A:
Let's assume the following for each racer:
-
Baby Einstein:
- Speed: 5 yards/second
- Starting distance from the finish line: 30 yards
Equation: \[ y = 5x + 30 \]
-
Toddler Newton:
- Speed: 3 yards/second
- Starting distance from the finish line: 25 yards
Equation: \[ y = 3x + 25 \]
-
Pre-teen Darwin:
- Speed: 4 yards/second
- Starting distance from the finish line: 20 yards
Equation: \[ y = 4x + 20 \]
Part B:
To determine when they will be the same distance from the finish line after a certain time, we can create a scenario. Let's assume that after 't' seconds, the distance they have traveled is the same.
Let's solve it assuming they all travel the same distance \(d\) yards in "t" seconds.
-
Baby Einstein: \[ y_{Einstein} = 5t + 30 \]
-
Toddler Newton: \[ y_{Newton} = 3t + 25 \]
-
Pre-teen Darwin: \[ y_{Darwin} = 4t + 20 \]
To find the time \(t\) and distance \(d\) at which their distances traveled are the same, we need to set the equations equal. For example, we can check when Baby Einstein and Toddler Newton are equal:
\[ 5t + 30 = 3t + 25 \] \[ 2t = -5 \] \[ t = -2.5 \quad (\text{not realistic in this context}) \]
Let’s assume the three finish the race together at the same time when they have traveled 15 yards from the starting point. That means:
- Baby Einstein's distance from the finish line: \[ 30 - 15 = 15 \text{ yards left} \]
- Toddler Newton's distance from the finish line: \[ 25 - 15 = 10 \text{ yards left} \]
- Pre-teen Darwin's distance from the finish line: \[ 20 - 15 = 5 \text{ yards left} \]
If you provide specific values for distance and time, I can adjust the calculations accordingly.