Question : If $h, C$, and $V$ are respectively the height, the curved surface, and the volume of a cone, then $3\pi Vh^{3} -C^{2}h^{2} + 9V^{2} =?$
Option 1: $0$
Option 2: $3$
Option 3: $\frac{1}{2}$
Option 4: $11$
Correct Answer: $0$
Solution :
For a cone of radius $r$, height $h$, and slant height $l$,
Volume, $V = \frac{1}{3}\pi r^{2}h$____(i)
Curved surface area, $C = \pi r l$____(ii)
$l ^2= r^{2} + h^{2}$____(iii)
Given expression,
$3\pi Vh^{3} -C^{2}h^{2} + 9V^{2}$
$=3\pi (\frac{1}{3}\pi r^{2}h)h^{3} -(\pi r l)^{2}h^{2} + 9( \frac{1}{3}\pi r^{2}h)^{2}$
$=\pi^2r^2h^4 -\pi^2r^2l^2h^2 +\pi^2r^4h^2$ (From equation (i) and (ii))
$=\pi^2r^2h^4 -\pi^2r^2(r^{2} + h^{2})h^2 +\pi^2r^4h^2$ (From equation (iii))
$=\pi^2r^2h^4-\pi^2r^2h^4+\pi^2r^4h^2-\pi^2r^4h^2$
$=0$
Hence, the correct answer is $0$.
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