Question 1178404
Let's break down this problem step-by-step.

**1. Visualize the Cone**

Imagine a cone representing the measuring glass.

* **Height (h):** 8 cm
* **Diameter (d):** 3 cm
* **Radius (r):** d/2 = 3/2 = 1.5 cm

**2. Volume of a Cone**

The volume (V) of a cone is given by:

* V = (1/3)πr²h

**3. Similar Cones**

When we fill the cone to different levels, we create smaller cones that are similar to the larger cone. This means their dimensions are proportional.

**4. Finding the Radii and Heights for 1 cc and 2 cc**

Let's denote:

* r1, h1: radius and height for 1 cc
* r2, h2: radius and height for 2 cc

We know:

* 1 = (1/3)πr1²h1
* 2 = (1/3)πr2²h2

Also, by similarity:

* r1/h1 = r/h = 1.5/8
* r2/h2 = r/h = 1.5/8

From these proportions, we have:

* r1 = (1.5/8)h1
* r2 = (1.5/8)h2

Substitute these into the volume equations:

* 1 = (1/3)π((1.5/8)h1)²h1 = (1/3)π(2.25/64)h1³
* 2 = (1/3)π((1.5/8)h2)²h2 = (1/3)π(2.25/64)h2³

Solve for h1 and h2:

* h1³ = (64 * 3) / (2.25π) ≈ 27.147
* h1 ≈ ∛27.147 ≈ 3.003 cm
* h2³ = (64 * 6) / (2.25π) ≈ 54.294
* h2 ≈ ∛54.294 ≈ 3.784 cm

Now find r1 and r2:

* r1 = (1.5/8) * 3.003 ≈ 0.563 cm
* r2 = (1.5/8) * 3.784 ≈ 0.710 cm

**5. Finding the Slant Edge Distances**

Let:

* s1: slant edge distance for 1 cc
* s2: slant edge distance for 2 cc

Using the Pythagorean theorem (in 3D):

* s1 = √(r1² + h1²) = √(0.563² + 3.003²) ≈ √(0.317 + 9.018) ≈ √9.335 ≈ 3.055 cm
* s2 = √(r2² + h2²) = √(0.710² + 3.784²) ≈ √(0.504 + 14.319) ≈ √14.823 ≈ 3.850 cm

**6. Finding the Distance Between Markings**

The distance on the slant edge between the 1 cc and 2 cc markings is:

* s2 - s1 ≈ 3.850 - 3.055 ≈ 0.795 cm

**Therefore, the distance on the slant edge between the markings for 1 cc and 2 cc is approximately 0.795 cm.**