A right circular cone resting on its base is cut at `(4)/(5)`th its height along a plane parallel to the circular base . The height of original cone is 75 cm and base diameter is 42 cm. What is the base radius of cut out top portion cone ?

Options

A. a. 4.2 cm
B. b. 2.8 cm
C. c. 3.5 cm
D. d. 8.4 cm

Answer

To solve the problem step by step, we will follow these steps:

### Step 1: Understand the dimensions of the original cone
The original cone has:
- Height (h) = 75 cm
- Base diameter = 42 cm

From the diameter, we can find the radius (r) of the base:
\[ r = \frac{\text{diameter}}{2} = \frac{42 \text{ cm}}{2} = 21 \text{ cm} \]

### Step 2: Determine the height at which the cone is cut
The cone is cut at \( \frac{4}{5} \) of its height. To find the height at which the cone is cut:
\[ \text{Height of cut portion} = \frac{4}{5} \times 75 \text{ cm} = 60 \text{ cm} \]

### Step 3: Find the height of the remaining cone
The height of the remaining cone (the part below the cut) can be calculated as:
\[ \text{Height of remaining cone} = \text{Total height} - \text{Height of cut portion} \]
\[ = 75 \text{ cm} - 60 \text{ cm} = 15 \text{ cm} \]

### Step 4: Use the similarity of triangles to find the radius of the cut-out cone
Since the cone is cut parallel to its base, the smaller cone formed (the cut-out top portion) is similar to the original cone. The ratio of the heights of the smaller cone to the original cone is:
\[ \text{Ratio of heights} = \frac{\text{Height of smaller cone}}{\text{Height of original cone}} = \frac{15 \text{ cm}}{75 \text{ cm}} = \frac{1}{5} \]

### Step 5: Calculate the radius of the base of the cut-out cone
Using the ratio of the heights, we can find the radius of the base of the cut-out cone:
\[ \text{Radius of cut-out cone} = \text{Radius of original cone} \times \text{Ratio of heights} \]
\[ = 21 \text{ cm} \times \frac{1}{5} = 4.2 \text{ cm} \]

### Final Answer
The base radius of the cut-out top portion cone is **4.2 cm**.

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