Directions: Read the information carefully and answer the following questions. A cube of side $C$ cm is placed inside the sphere of radius $R$ cm in such a way that sphere touches all the vertex of the cube. A cone of radius $\sqrt{3}R$ cm and height $H$ cm has volume $3168 \text{ cm}^3$. What is the relation between the $C$ and $H$?

### Concept & Algebraic Substitution in Geometry By finding a common variable (in this case, the sphere's radius $R$) between the cube and the cone, we can link the given volume of the cone directly to the dimensional side of the cube. $$ \text{Volume of Cone} = \frac{1}{3}\pi r^2 h $$ ### Step-by-Step Solution 1. **Analyze the Cone's Volume:** - Radius of cone = $\sqrt{3}R$ - Height of cone = $H$ - Volume = $3168 \text{ cm}^3$ - $\frac{1}{3} \times \pi \times (\sqrt{3}R)^2 \times H = 3168$ 2. **Simplify the Cone Equation:** - $\frac{1}{3} \times \frac{22}{7} \times 3R^2 \times H = 3168$ - The 3s cancel out: $\frac{22}{7} \times R^2 \times H = 3168$ - $R^2H = 3168 \times \frac{7}{22}$ - $3168 / 22 = 144$. So, $R^2H = 144 \times 7 = 1008$. 3. **Relate $R$ back to $C$:** - From the initial conditions (cube inscribed in sphere), the body diagonal equals the diameter: $\sqrt{3}C = 2R$. - Squaring both sides: $3C^2 = 4R^2 \Rightarrow R^2 = \frac{3C^2}{4}$. 4. **Substitute and Solve for Relation:** - Substitute $R^2$ into the simplified volume equation: $(\frac{3C^2}{4}) \times H = 1008$ - $3C^2H = 1008 \times 4$ - $3C^2H = 4032$ - $C^2H = \frac{4032}{3} = 1344$ - $C^2 = \frac{1344}{H}$ ### Exam Strategy & Shortcut Rather than calculating $144 \times 7$ immediately, keep the terms in factored form as $144 \times 7 \times \frac{4}{3}$ when bridging the two formulas. $144 / 3 = 48$, and $48 \times 28 = 1344$. Delayed multiplication reduces arithmetic errors and speeds up the simplification process. ### Common Pitfall Forgetting to properly square the $\sqrt{3}$ coefficient when substituting the radius of the cone into the volume formula, which fundamentally alters the entire factor chain. ### Final Answer Therefore, the correct answer is **$C^2 = \frac{1344}{H}$**.