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

Step 1: Use \( v = \frac{1}{3}\pi r^2 h \) and \( c = \pi r l \) with \( l^2 = r^2+h^2 \).

Step 2: \( 9v^2 = \pi^2 r^4 h^2 \), and \( c^2h^2 = \pi^2 r^2(r^2+h^2)h^2 = \pi^2 r^4h^2 + \pi^2 r^2h^4 \).

Step 3: Subtracting, \( c^2h^2 - 9v^2 = \pi^2 r^2 h^4 \).

Step 4: Directly, \( 3\pi v h^3 = 3\pi \left(\frac{1}{3}\pi r^2h\right)h^3 = \pi^2r^2h^4 \), the same expression. \[ \boxed{3\pi v h^3 = c^2h^2 - 9v^2} \]