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 ratio of the total surface area of the sphere to the lateral surface area of the cube?

### Concept & Mensuration of 3D Figures When a cube is inscribed inside a sphere (touching all vertices), the body diagonal of the cube is exactly equal to the diameter of the sphere. $$ \text{Body Diagonal} = \sqrt{3} \times C $$ $$ \text{Diameter} = 2 \times R $$ ### Step-by-Step Solution 1. **Establish the relation between $C$ and $R$:** - Since the sphere touches all vertices of the cube, the longest distance inside the cube (body diagonal) spans across the sphere's center. - $\sqrt{3}C = 2R$ - Squaring both sides: $3C^2 = 4R^2 \Rightarrow C^2 = \frac{4}{3}R^2$ 2. **Calculate Total Surface Area (TSA) of the sphere:** - $\text{TSA}_{\text{sphere}} = 4\pi R^2 = 4 \times \frac{22}{7} \times R^2 = \frac{88}{7}R^2$ 3. **Calculate Lateral Surface Area (LSA) of the cube:** - The lateral surface area comprises 4 vertical faces. - $\text{LSA}_{\text{cube}} = 4C^2$ - Substituting $C^2 = \frac{4}{3}R^2$: $\text{LSA}_{\text{cube}} = 4 \times (\frac{4}{3}R^2) = \frac{16}{3}R^2$ 4. **Find the Ratio:** - $\text{Ratio} = \frac{\text{TSA}_{\text{sphere}}}{\text{LSA}_{\text{cube}}} = \frac{\frac{88}{7}R^2}{\frac{16}{3}R^2}$ - $\text{Ratio} = \frac{88}{7} \times \frac{3}{16} = \frac{11}{7} \times \frac{3}{2} = \frac{33}{14}$ ### Exam Strategy & Shortcut Always leave $\pi$ as $22/7$ when ratios are involved because standard options will often perfectly cancel out common factors like 11 or 4. Also, write out formulas symbolically before plugging any values in to avoid redundant or nested calculations. ### Common Pitfall A very common mistake is confusing Total Surface Area (6 faces) with Lateral Surface Area (4 faces) of the cube. The question explicitly asks for the lateral surface area ($4C^2$). ### Final Answer Therefore, the correct answer is **33 : 14**.