Ratio of the height of the cone and cylinder is $3 : 2$. Ratio of the radius of the cone and cylinder is $3 : 4$. Ratio of the volume of the cone and cylinder is $m : n$. The length of the cuboid is $(m + 11)$ and the breadth of the cuboid is $(n - 11)$. The total volume of the cuboid is 11340. Find the possible value of the height of the cuboid? I. $3m$ II. $\frac{n}{4}$ III. $n - m - 15$

### Concept & Volume Proportions The key is establishing the relationship between the volumes of a cone and a cylinder using their given dimension ratios. The volume of a cone is $V_{\text{cone}} = \frac{1}{3}\pi r^2 h$ and the volume of a cylinder is $V_{\text{cylinder}} = \pi r^2 h$. The ratio of their volumes is: $$ \frac{V_{\text{cone}}}{V_{\text{cylinder}}} = \frac{\frac{1}{3}\pi r_1^2 h_1}{\pi r_2^2 h_2} $$ ### Step-by-Step Solution * **Establish Ratios:** Let the radii of the cone and cylinder be $3x$ and $4x$ respectively. Let the heights of the cone and cylinder be $3y$ and $2y$ respectively. * **Calculate Volume Ratio ($m : n$):** $$ \frac{m}{n} = \frac{\frac{1}{3} \pi (3x)^2 (3y)}{\pi (4x)^2 (2y)} $$ $$ \frac{m}{n} = \frac{1}{3} \times \frac{9x^2}{16x^2} \times \frac{3y}{2y} $$ $$ \frac{m}{n} = \frac{1}{3} \times \frac{9}{16} \times \frac{3}{2} = \frac{9}{32} $$ Assuming the simplest integer values for the ratio, $m = 9$ and $n = 32$. * **Calculate Cuboid Dimensions:** Length of cuboid $L = m + 11 = 9 + 11 = 20$. Breadth of cuboid $B = n - 11 = 32 - 11 = 21$. * **Find Height of Cuboid ($H$):** Total volume of cuboid = $L \times B \times H = 11340$. $$ 20 \times 21 \times H = 11340 $$ $$ 420 \times H = 11340 \implies H = 27 $$ * **Evaluate Statements:** I. $3m = 3(9) = 27$ (Matches) II. $\frac{n}{4} = \frac{32}{4} = 8$ (Does not match) III. $n - m - 15 = 32 - 9 - 15 = 8$ (Does not match) Only statement I yields the correct height. ### Exam Strategy & Shortcut When variables expressed directly from ratios (like $m$ and $n$ forming $m+11$) are used as absolute lengths, it strongly implies that the base simplified ratio values (9 and 32) are the actual intended absolute values. Plug $k=1$ in immediately to save time, and quickly verify if the resulting numbers divide cleanly (like $11340 \div 420 = 27$). ### Common Pitfall Forgetting the $\frac{1}{3}$ factor in the cone's volume formula, which would drastically alter the $m:n$ ratio and lead to incorrect cuboid dimensions. ### Final Answer Therefore, the correct answer is **Only I**.