The given question is followed by three statements labelled I, II and III. You have to study the question and all the three statements given do decide whether any information provided in the statement(s) is/are redundant and can be dispensed with while answering the given question.
A solid metallic cone is melted and recast into a sphere. What is the radius of the sphere?
I. The radius of the base of the cone is 2.1 cm.
II. The height of the cone is four times the radius of its base.
III. The height of the cone is 8.4 cm.

Options

A. Only I and II
B. Only I and III
C. Both a and b
D. Any two of the three

Answer

To solve the problem of finding the radius of the sphere formed by melting a solid metallic cone, we will analyze the given statements and determine which ones are necessary for finding the radius.

### Step-by-Step Solution:

1. **Understanding the Volume Relationship**:
The volume of the cone is equal to the volume of the sphere since the cone is melted and recast into the sphere.
\[ V_{cone} = V_{sphere} \]

2. **Volume Formulas**:
The volume of a cone is given by:
\[ V_{cone} = \frac{1}{3} \pi r^2 h \]
where \( r \) is the radius of the base of the cone and \( h \) is the height of the cone.

The volume of a sphere is given by:
\[ V_{sphere} = \frac{4}{3} \pi R^3 \]
where \( R \) is the radius of the sphere.

3. **Setting the Volumes Equal**:
Equating the two volumes:
\[ \frac{1}{3} \pi r^2 h = \frac{4}{3} \pi R^3 \]
We can simplify this by canceling \(\pi\) and \(\frac{1}{3}\):
\[ r^2 h = 4R^3 \]

4. **Analyzing the Statements**:
- **Statement I**: The radius of the base of the cone is 2.1 cm.
- **Statement II**: The height of the cone is four times the radius of its base.
- **Statement III**: The height of the cone is 8.4 cm.

5. **Using the Statements**:
- From **Statement I**, we know \( r = 2.1 \) cm.
- From **Statement II**, if \( r = 2.1 \) cm, then \( h = 4 \times 2.1 = 8.4 \) cm.
- **Statement III** directly gives us \( h = 8.4 \) cm.

6. **Redundancy Check**:
- **Statement I** is necessary to find \( r \).
- **Statement II** provides a relationship that allows us to find \( h \) if we know \( r \).
- **Statement III** gives us \( h \) directly.

If we have **Statement I** and **Statement II**, we can find both \( r \) and \( h \), which allows us to calculate the radius of the sphere. However, if we have **Statement I** and **Statement III**, we can also find \( r \) and \( h \).

Therefore, **Statement II** is redundant because we can derive the height from the other two statements.

7. **Conclusion**:
We can conclude that **Statement II** is redundant. Thus, we can answer the question with either **Statements I and III** or **Statements I and II**.

### Final Answer:
The redundant statement is **Statement II**.

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