If $$w_1$$​ is the weight with cargo of a ship of length $$l_1$$​, and $$w_2$$​ and $$w_3$$​ are the corresponding weights of ships having lengths $$l_2$$​ and $$l_3$$​ respectively, then

​$$\left\{\frac{w_1}{l_1{ }^2}\right\}\left(l_2-l_3\right)+\left\{\frac{w_2}{l_2}\right\}\left(l_3-l_1\right)+\left\{\frac{w_3}{l_3{ }^2}\right\}\left(l_1-l_2\right)=0 .$$​​

Which one of the following is correct in respect of the Question and the Statements given below?

Statement 1: The weight of an empty ship varies as the square of the length of the ship.

Statement 2: The weight of the ship's cargo varies as the cube of the length of the ship.

Statement 3: The weight of the ship with cargo varies as the sixth power of the length of the ship.

Correct option is C

​$$\textbf{Solution:}\\\quad \text{Given:} \\\quad \left( \frac{w_1}{l_1^2} \right) (l_2 - l_3) + \left( \frac{w_2}{l_2} \right) (l_3 - l_1) + \left( \frac{w_3}{l_3^2} \right) (l_1 - l_2) = 0\\\quad \textbf{Step 1:} \\\quad \text{Observe the terms: } \left( \frac{w_1}{l_1^2} \right), \left( \frac{w_2}{l_2} \right), \text{ and } \left( \frac{w_3}{l_3^2} \right). \\\ \\\quad \text{Each term shows a different dependence on length.}\\\quad \textbf{Step 2:} \\\quad \text{Statement 1: Weight of an empty ship varies as the square of the length, i.e., } w_{\text{empty}} \propto l^2.\\\quad \text{Statement 2: Weight of the ship's cargo varies as the cube of the length, i.e., } w_{\text{cargo}} \propto l^3.\\\quad \textbf{Step 3:} \\\quad \text{Thus, total weight of the ship is given by } w_{\text{total}} = w_{\text{empty}} + w_{\text{cargo}}. \\\quad \text{This means weight depends on both } l^2 \text{ and } l^3 \text{ terms.}\\\quad \textbf{Step 4:} \\\quad \text{Statement 3 says weight varies as sixth power of length.} \\\quad \text{This is incorrect because sixth power would occur only if terms were multiplied, not added.}\\\quad \textbf{Conclusion:} \\\quad \text{Thus, Statement 1 and Statement 2 together are sufficient to satisfy the given relation.}$$

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