Cone

A circular cone has height h = 15 dm and base radius r = 2 dm. A plane parallel to the base divides the cone into two parts of equal volume. Calculate the distance from the cone's vertex to this plane.

Final Answer:

x =  11.9 dm

Step-by-step explanation:

$$\begin{gathered} h=15\text{dm} \\ r=2\text{dm} \\ {r}_1:x=r:h \\ V=\frac{1}{3}\cdot \pi \cdot r^2\cdot h=\frac{1}{3}\cdot 3.1416\cdot 2^2\cdot 15\doteq 62.8319{\text{dm}}^3 \\ {V}_1=V_2;V_1+V_2=V \\ {V}_1=\frac{1}{2}\cdot V=\frac{1}{2}\cdot 62.8319\doteq 31.4159{\text{dm}}^3 \\ {V}_1=\frac{1}{3}\cdot \pi \cdot r_1^2\cdot x \\ {V}_1=\frac{1}{3}\cdot \pi \cdot (r/h\cdot x{)}^2\cdot x \\ {V}_1=\frac{1}{3}\cdot \pi \cdot r^2/h^2\cdot x^3 \\ x=\sqrt[ \\ 3]{\frac{3\cdot V_1\cdot h^2}{\pi \cdot r^2}}=\sqrt[ \\ 3]{\frac{3\cdot 31.4159\cdot 15^2}{3.1416\cdot 2^2}}=11.9\text{dm} \end{gathered}$$

Try another example