Step-by-step explanation:
[tex]\texttt{Volume of cone, V = }\frac{1}{3}\pi r^2h[/tex]
For the first cone
Height, h = 18
Radius, r = 12
Substituting
[tex]\texttt{Volume of cone, V = }\frac{1}{3}\pi r^2h\\\\\texttt{Volume of cone, V = }\frac{1}{3}\times \pi \times 12^2\times 18=2714.34[/tex]
Volume of new cone formed is half of the older cone.
Volume of new cone = 0.5 x 2714.34 = 1357.17
For the cone as the height reduces to 18 radius reduces to zero.
[tex]tan\theta =\frac{12}{18}\\\\\theta =33.7^0[/tex]
[tex]\texttt{Volume of new cone = }\frac{1}{3}\pi r_1^2h_1\\\\tan\theta =\frac{r_1}{h_1}\Rightarrow r_1=h_1tan\theta\Rightarrow r_1=h_1tan33.7=0.667h_1\\\\\texttt{Volume of new cone = }\frac{1}{3}\pi \times (0.667h_1)^2h_1=0.467h_1^3[/tex]
We have
0.467h₁³ = 1357.17
h₁ = 14.29
Height of smaller cone = 14.29
