Question

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=

Toddler Newton: y=
Pre-teen Darwin: y=


Part B: Baby Einstein, Toddler Newton, and Pre-teen Darwin will be the same distance from the finish line after
seconds and the distance they all have traveled is
yards.

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Answer 1

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:**
- **Baby Einstein** starts with a head start of 5 yards and runs at a speed of 3 yards per second.
\[
y = 3x + 5
\]

- **Toddler Newton** starts with a head start of 10 yards and runs at a speed of 2 yards per second.
\[
y = 2x + 10
\]

- **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.

1. **Set Baby Einstein equal to Toddler Newton:**
\[
3x + 5 = 2x + 10
\]
\[
3x - 2x = 10 - 5
\]
\[
x = 5 \text{ seconds}
\]

2. **Set Toddler Newton equal to Pre-teen Darwin:**
\[
2x + 10 = 4x
\]
\[
10 = 4x - 2x
\]
\[
2x = 10
\]
\[
x = 5 \text{ seconds}
\]

3. **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:
- For **Baby Einstein**:
\( y = 3(5) + 5 = 15 + 5 = 20 \text{ yards} \)

- For **Toddler Newton**:
\( y = 2(5) + 10 = 10 + 10 = 20 \text{ yards} \)

- For **Pre-teen Darwin**:
\( y = 4(5) = 20 \text{ yards} \)

**Final Answers:**
- **Part A:**
- Baby Einstein: \( y = 3x + 5 \)
- Toddler Newton: \( y = 2x + 10 \)
- Pre-teen Darwin: \( y = 4x \)

- **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!