A satellite is currently in a circular orbit with radius 32,714 km about the center of the earth.

Phase 1: Owning the Dataset

Visualization

Let's draw the satellite's circular orbit:

        Earth (center)
             ●
            /|\
           / | \
    r = 32,714 km
         /   |   \
        ○----+----○  Circular orbit
         \   |   /   Distance per revolution = 2πr
          \  |  /
           \ | /
            \|/

Key relationships:

• Distance traveled in one revolution = Circumference = \(2\pi r\)
• Current orbit radius: \(r_1 = 32,714\) km
• If radius increases by \(\Delta r\), new radius = \(r_1 + \Delta r\)

Phase 2: Understanding the Question

What we're asked to find:

• Part 1 (0.5 km): Increase in distance when radius increases by 0.5 km
• Part 2 (1.5 km): Increase in distance when radius increases by 1.5 km

Key Insight

When the radius increases from r to (r + Δr), the increase in circumference is:

• New circumference: \(2\pi(r + \Delta r)\)
• Old circumference: \(2\pi r\)
• Increase = \(2\pi(r + \Delta r) - 2\pi r = 2\pi\Delta r\)

Notice: The increase depends ONLY on \(\Delta r\), not on the original radius!

Phase 3: Finding the Answer

Calculations

For 0.5 km radius increase:

• Increase in distance = \(2\pi(0.5) = \pi\) km
• \(\pi \approx 3.14159...\) km
• To nearest whole kilometer: 3 km ✓

For 1.5 km radius increase:

• Increase in distance = \(2\pi(1.5) = 3\pi\) km
• \(3\pi \approx 9.42478...\) km
• To nearest whole kilometer: 9 km ✓

Verification with Answer Choices

Our answer choices are: [3km, 9km, 27km, 81km, 150km, 450km]

• 0.5 km → 3 km ✓ (matches first choice)
• 1.5 km → 9 km ✓ (matches second choice)

Phase 4: Solution

Final Answer:

• 0.5 km: 3km
• 1.5 km: 9km

The elegant mathematical principle here is that the increase in orbital distance is simply \(2\pi\) times the increase in radius, regardless of the initial orbit size. This makes our calculation beautifully simple!