Question

If \( h \), \( c \), \( v \) are the height, the curved surface area, and the volume of a cone, then \( 3\pi vh^3 \) is:

• A. \( 9v^2 - c^2h^3 \)
• B. \( c^2h^2 - 9v^2 \)
• C. \( c^2h^2 - 9v^3 \)
• D. \( c^2h^2 - 16v^2 \)

Answer 1

The Correct Option is B

Given:
- \( h \): Height of the cone
- \( c \): Curved surface area of the cone
- \( v \): Volume of the cone
We need to evaluate the expression \( 3\pi vh^3 \).
Let's analyze each option and find the correct one:
Options:
A) \( 9v^2 - c^2h^3 \)
B) \( c^2h^2 - 9v^2 \)  
C) \( c^2h^2 - 9v^3 \)  
D) \( c^2h^2 - 16v^2 \)
Solution:
1. Expression Analysis:
  \( 3\pi vh^3 \) represents the given expression.
2. Evaluation of Options:
- Option A: \( 9v^2 - c^2h^3 \)
    - This option does not match \( 3\pi vh^3 \).
- Option B: \( c^2h^2 - 9v^2 \)
    - Here, \( c^2h^2 \) corresponds to the square of the curved surface area times the square of the height of the cone.
    - \( 9v^2 \) represents 9 times the square of the volume of the cone.
    - This option matches the form \( x - y \), where \( x = c^2h^2 \) and \( y = 9v^2 \).
- Option C: \( c^2h^2 - 9v^3 \)
    - This option does not match \( 3\pi vh^3 \).
- Option D: \( c^2h^2 - 16v^2 \)
    - This option also does not match \( 3\pi vh^3 \).
3. Conclusion:
  After evaluating all options, the expression \( 3\pi vh^3 \) matches with Option B:
Option B) \( c^2h^2 - 9v^2 \)
Therefore, the correct answer is Option B). This option correctly represents \( 3\pi vh^3 \) in the given format \( x - y \), where \( x = c^2h^2 \) and \( y = 9v^2 \).

Answer 2

Rather than testing each option against the target expression, derive \( 3\pi v h^3 \) directly from the standard cone formulas and see which option it matches exactly.

1. Step 1: For a cone of base radius \( r \), height \( h \) and slant height \( l \), the volume is \[ v = \frac{1}{3}\pi r^2 h \] and the curved surface area is \[ c = \pi r l, \quad \text{where } l^2 = r^2 + h^2 \]
2. Step 2: Square the volume formula: \[ v^2 = \frac{1}{9}\pi^2 r^4 h^2 \implies 9v^2 = \pi^2 r^4 h^2 \]
3. Step 3: Square the curved surface area formula: \[ c^2 = \pi^2 r^2 l^2 = \pi^2 r^2 (r^2 + h^2) = \pi^2 r^4 + \pi^2 r^2 h^2 \] Multiplying through by \( h^2 \): \[ c^2 h^2 = \pi^2 r^4 h^2 + \pi^2 r^2 h^4 \]
4. Step 4: Subtract \( 9v^2 \) from \( c^2h^2 \): \[ c^2h^2 - 9v^2 = \left(\pi^2 r^4 h^2 + \pi^2 r^2 h^4\right) - \pi^2 r^4 h^2 = \pi^2 r^2 h^4 \]
5. Step 5: Now evaluate \( 3\pi v h^3 \) directly: \[ 3\pi v h^3 = 3\pi \left(\frac{1}{3}\pi r^2 h\right) h^3 = \pi^2 r^2 h^4 \]

Both expressions reduce to exactly \( \pi^2 r^2 h^4 \), so \( 3\pi v h^3 = c^2h^2 - 9v^2 \) precisely, with nothing left over.

Therefore, the correct answer is \( c^2h^2 - 9v^2 \).