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

Answer: (B)

$c^2h^2 - 9v^2$

Answer 2

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 $y = 9v^2$.