Question

a right circular cone has a height of 18 and a base radius of 12. A slice parallel to the base in made completely throught the cone, and the resulting smaller cone has a volume that is 1/2 the volume of the original cone. What is the height of the smaller cone?

Answer 1

Height of smaller cone is 14.29

Step-by-step explanation:

$\texttt{Volume of cone, V = }\frac{1}{3}\pi r^2h$

For the first cone

            Height, h = 18

            Radius, r = 12

Substituting

           $\texttt{Volume of cone, V = }\frac{1}{3}\pi r^2h\\\\\texttt{Volume of cone, V = }\frac{1}{3}\times \pi \times 12^2\times 18=2714.34$

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.

               $tan\theta =\frac{12}{18}\\\\\theta =33.7^0$

  $\texttt{Volume of new cone = }\frac{1}{3}\pi r_1^2h_1\\\\tan\theta =\frac{r_1}{h_1}\Rightarrow r_1=h_1tan\theta\Rightarrow r_1=h_1tan33.7=0.667h_1\\\\\texttt{Volume of new cone = }\frac{1}{3}\pi \times (0.667h_1)^2h_1=0.467h_1^3$

We have

              0.467h₁³ = 1357.17

                       h₁ = 14.29

Height of smaller cone = 14.29