Question

A cone of radius 9 cm and height 15 cm is melted and made into a cylinder of height 45 cm. The diameter of the cylinder is:

• A. 3 cm
• B. 5 cm
• C. 10 cm
• D. 6 cm

Hint:

1. When equating volumes of shapes, keep the constant \( \pi \) as it is because it will cancel out on both sides, saving time and preventing calculation mistakes.
2. Always read the final question carefully: many candidates make the mistake of choosing the radius (3 cm) instead of the diameter (6 cm).
3. Knowing basic factors (like \( 45 \times 9 = 405 \)) makes mental division extremely fast.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
1. This question deals with the conservation of volume during the process of melting and reshaping a solid.
2. We are given the dimensions of a solid cone (radius and height) and the height of the newly formed cylinder.
3. The objective is to find the diameter of this resulting cylinder.
4. Since the cone is melted and completely reshaped into a cylinder with no loss of material, the volume of the cone must equal the volume of the cylinder.

Step 2: Key Formula or Approach:
1. The formula for the volume of a right circular cone is:
\[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \]
where \( r \) is the radius of the cone's base and \( h \) is its vertical height.
2. The formula for the volume of a cylinder is:
\[ V_{\text{cylinder}} = \pi R^2 H \]
where \( R \) is the radius of the cylinder's base and \( H \) is its height.
3. Since the volumes are equal:
\[ V_{\text{cone}} = V_{\text{cylinder}} \]
4. Once the radius \( R \) is found, the diameter \( D \) is calculated as:
\[ D = 2R \]

Step 3: Detailed Explanation:
1. Firstly, let us record the given dimensions of the cone:
- Radius of the cone, \( r = 9 \text{ cm} \)
- Height of the cone, \( h = 15 \text{ cm} \)
2. Secondly, we substitute these dimensions into the cone volume formula:
\[ V_{\text{cone}} = \frac{1}{3} \pi (9)^2 (15) \]
\[ V_{\text{cone}} = \frac{1}{3} \pi (81) (15) \]
3. Simplifying the expression:
\[ V_{\text{cone}} = \pi (81) (5) = 405\pi \text{ cm}^3 \]
4. Thirdly, let us denote the unknown radius of the cylinder as \( R \).
5. The given height of the cylinder is \( H = 45 \text{ cm} \).
6. The volume of this cylinder is:
\[ V_{\text{cylinder}} = \pi R^2 (45) = 45\pi R^2 \text{ cm}^3 \]
7. Fourthly, since the cone is converted into the cylinder, we equate their volumes:
\[ 45\pi R^2 = 405\pi \]
8. We can divide both sides by \( \pi \):
\[ 45 R^2 = 405 \]
9. Now, solving for \( R^2 \) by dividing both sides by 45:
\[ R^2 = \frac{405}{45} = 9 \]
10. Taking the square root of both sides to find the radius \( R \):
\[ R = \sqrt{9} = 3 \text{ cm} \]
11. Fifthly, the question asks for the diameter of the cylinder, not the radius.
12. The diameter \( D \) of a cylinder is twice its radius:
\[ D = 2R = 2(3) = 6 \text{ cm} \]

Step 4: Final Answer:
1. The diameter of the resulting cylinder is determined to be 6 cm.
2. Therefore, this matches Option (D).