2x cone

Circular cone height 36 cm was cut plane parallel with the base. The volume of these two small cones is the same. Calculate the height of the smaller cone.

Final Answer:

v₂ =  28.5732 cm

Step-by-step explanation:

$$\begin{gathered} v_1=36\text{cm} \\ {r}_1=1\text{cm} \\ {V}_1=\frac{1}{3}\cdot \pi \cdot r_1^2\cdot v_1=\frac{1}{3}\cdot 3.1416\cdot 1^2\cdot 36\doteq 37.6991{\text{cm}}^3 \\ {V}_2=V_1/2=37.6991/2\doteq 18.8496{\text{cm}}^3 \\ {r}_1:v_1=r_2:v_2 \\ {V}_2=\frac{1}{3}\cdot \pi \cdot r_2^2\cdot v_2 \\ {V}_2=\frac{1}{3}\cdot \pi \cdot (r_1/v_1\cdot v_2{)}^2\cdot v_2 \\ {V}_2=\frac{1}{3}\cdot \pi \cdot r_1^2/v_1^2\cdot v_2^3 \\ {v}_2=\sqrt[3]{3\cdot V_2/\pi /r_1^2\cdot v_1^2}=\sqrt[3]{3\cdot 18.8496/3.1416/1^2\cdot 36^2}\doteq 28.5732\text{cm} \\ {r}_2=r_1/v_1\cdot v_2=1/36\cdot 28.5732\doteq 0.7937\text{cm} \\ {V}_3=\frac{1}{3}\cdot \pi \cdot r_2^2\cdot v_2=\frac{1}{3}\cdot 3.1416\cdot 0.7937^2\cdot 28.5732\doteq 18.8496{\text{cm}}^3 \\ {V}_3=V_2=V_1/2 \end{gathered}$$

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