The height and the radius of the base of a right circular cone are 12 cm and 6 cm respectively. The radius of the circular cross-section of the cone cut by a plane parallel to its base at a distance of 3 cm from the base is

Options

A. 4 cm
B. 5.5 cm
C. 4.5 cm
D. 3.5 cm

Answer

To find the radius of the circular cross-section of a cone cut by a plane parallel to its base, we can use the concept of similar triangles. Here’s a step-by-step solution:

### Step 1: Understand the dimensions of the cone
We have a right circular cone with:
- Height (h) = 12 cm
- Radius of the base (r) = 6 cm

### Step 2: Identify the position of the cut
The plane cuts the cone parallel to its base at a distance of 3 cm from the base. This means the height of the smaller cone (above the cut) is:
- Height of the smaller cone (h1) = Total height - Distance from the base = 12 cm - 3 cm = 9 cm

### Step 3: Set up the ratio of the dimensions
Since the triangles formed by the cone and the smaller cone are similar, we can set up a ratio of their corresponding dimensions (heights and radii).

Let:
- r1 = radius of the circular cross-section at the height of 9 cm (which we need to find)
- h1 = height of the smaller cone = 9 cm
- h = total height of the cone = 12 cm
- r = radius of the base of the cone = 6 cm

Using the property of similar triangles:
\[ \frac{r1}{r} = \frac{h1}{h} \]

### Step 4: Substitute the known values
Substituting the known values into the ratio:
\[ \frac{r1}{6} = \frac{9}{12} \]

### Step 5: Simplify the ratio
Now, simplify the right side:
\[ \frac{9}{12} = \frac{3}{4} \]

So we have:
\[ \frac{r1}{6} = \frac{3}{4} \]

### Step 6: Solve for r1
Cross-multiply to find r1:
\[ 4r1 = 6 \times 3 \]
\[ 4r1 = 18 \]
\[ r1 = \frac{18}{4} = 4.5 \text{ cm} \]

### Conclusion
The radius of the circular cross-section of the cone cut by a plane parallel to its base at a distance of 3 cm from the base is **4.5 cm**.

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