Cone

The circular cone has height h = 15 dm and base radius r = 2 dm slice plane parallel to the base. Calculate the distance of the cone vertex from this plane if solids have the same volume.

Correct answer:

x =  112.5 dm

Step-by-step explanation:

$$\begin{gathered} h=15\text{dm} \\ r=2\text{dm} \\ {V}_1=V_2 \\ {r}_1:r=x:h \\ {V}_1=\frac{1}{3}\cdot \pi \cdot r_1^2\cdot x=\frac{1}{3}\cdot \pi \cdot (x\cdot r/h{)}^2\cdot x \\ {V}_2=\frac{1}{2}\cdot \frac{1}{3}\cdot \pi \cdot r^2\cdot h \\ \frac{1}{3}\pi (x\cdot r/h{)}^2=\frac{1}{6}\pi r^2\cdot x \\ \frac{1}{3}(x\cdot /h{)}^2=\frac{1}{6}\cdot x \\ {x}^2/h^2=\frac{1}{2}\cdot x \\ x/h^2=\frac{1}{2} \\ x=\frac{h^2}{2}=\frac{15^2}{2}=112.5\text{dm} \end{gathered}$$

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