Questions: Cantaloupe, radii 4 and 6, 30 degree slice. 4 ≤ ρ ≤ 6 0 ≤ φ ≤ π 0 ≤ θ ≤ π/6 √(16-r^2) ≤ z ≤ √(36-r^2) 4 ≤ r ≤ 6 0 ≤ θ ≤ π/6 4 ≤ z ≤ 6 0 ≤ φ ≤ π/2 0 ≤ θ ≤ π/6

Solution Steps

We are asked to determine the bounds of a cantaloupe slice in spherical coordinates. The cantaloupe has inner radius 4 and outer radius 6. The slice covers 30 degrees, or π/6 radians.

The radial distance, ρ, goes from the inner radius to the outer radius. Thus, 4 ≤ ρ ≤ 6.

The angle Φ sweeps down from the z-axis. The image shows that we have a full rotation, meaning Φ goes from 0 to π. So, 0 ≤ Φ ≤ π.

The angle θ is given as a 30 degree slice (π/6 radians). From the bottom images, we can see that θ sweeps from the y-axis to the slice. This means the angle starts at zero and sweeps to π/6. Thus, 0 ≤ θ ≤ π/6.

Final Answer: