A portion of an automobile test track is divided into Segment A, Segment B, and Segment C, in that order....

Phase 1: Owning the Dataset

Visual Representation

Let's draw a timeline showing the car's journey:

Segment A -----> Segment B -----> Segment C
140 km/h         (slowing)        70 km/h
length: x        (ignored)        length: 3x

Key relationships:

• Speed in A: 140 km/h (constant)
• Speed in C: 70 km/h (constant)
• Length of C = \(3 \times \mathrm{Length\ of\ A}\)
• Total time for A and C: 42 minutes
• Segment B is ignored (no time/distance given)

Phase 2: Understanding the Question

Setting Up Variables

Let's define:

• Length of Segment A = \(x\) kilometers
• Length of Segment C = \(3x\) kilometers (given relationship)

Time Calculations

For constant speed segments, we use: \(\mathrm{Time} = \mathrm{Distance} \div \mathrm{Speed}\)

• Time for Segment A = \(x \div 140\) hours
• Time for Segment C = \(3x \div 70\) hours
• Total time = 42 minutes = \(\frac{42}{60} = 0.7\) hours

Phase 3: Finding the Answer

Setting Up the Equation

Total time for A and C:

\(\frac{x}{140} + \frac{3x}{70} = 0.7\)

Solving Step by Step

Let's find a common denominator:

\(\frac{x}{140} + \frac{3x}{70} = 0.7\)

\(\frac{x}{140} + \frac{6x}{140} = 0.7\) (since \(\frac{3x}{70} = \frac{6x}{140}\))

\(\frac{7x}{140} = 0.7\)

\(\frac{x}{20} = 0.7\)

\(x = 14\)

Therefore:

• Length of Segment A = 14 km
• Length of Segment C = \(3 \times 14 = 42\) km

Verification

Let's check our answer:

• Time for A: \(14 \div 140 = 0.1\) hours = 6 minutes
• Time for C: \(42 \div 70 = 0.6\) hours = 36 minutes
• Total: \(6 + 36 = 42\) minutes ✓

Phase 4: Solution

Final Answer:

• Length of Segment A: 14 kilometers
• Length of Segment C: 42 kilometers

Our answer satisfies all given conditions: Segment C is 3 times longer than Segment A (\(42 = 3 \times 14\)), and the total travel time for both segments equals exactly 42 minutes.