Given:
The radius of the cone is given as r.
The height of the cone is given as h = 3r.
The objective is to find the volume of the cone.
Explanation:
The general formula for the volume of a cone is,
[tex]V=\frac{1}{3}\pi r^2h\text{ . . . . . .(1)}[/tex]
Substitute the value of h in the equation (1),
[tex]\begin{gathered} V=\frac{1}{3}\pi r^2(3r) \\ V=\pi r^3\text{ . . . . . .(2)} \end{gathered}[/tex]
On solving the equation of height given in the figure in terms of radius,
[tex]\begin{gathered} h=3r \\ r=\frac{h}{3}\text{ . . . . .(3)} \end{gathered}[/tex]
Substitute the equation (3) in equation (2),
[tex]\begin{gathered} V=\pi(\frac{h}{3})^3 \\ V=\pi\frac{h^3}{27}\text{ . . . . . .(4)} \end{gathered}[/tex]
Thus, both equation (2) and equation (4) are the volumes of the cone.
Hence, options (1) and (3) are the correct answers.
