Question related two different ways to do solution

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The discussion revolves around the different methods used to solve a problem involving cross-section sidelengths. One method expresses the sidelength as 8-8x/3, while the other relies on similar triangle relationships. The user questions the correctness of both approaches and seeks clarification on which answer is accurate, either 470 kJ or 117 kJ. There is also a suggestion to try the integral method indicated in red. The conversation highlights the complexities of solving the problem and the potential for multiple valid approaches.
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Homework Statement
An upside-down square pyramid with a height of 3 m and side lengths of 8m is filled to the brim with water. How much work is needed to lift all the water out of the container through the top. (ρ = 1000 kg/m3, g = 9.8 m/s2)
Relevant Equations
Work = Force * displacement
My work:
Screenshot 2025-04-12 203210.png


Answer Key (red font one) *may not be right.

My question is why is sidelength of cross section written as 8-8x/3, while my way of doing it doesn't involve that (involves suspicious similar triangle relationships)? Which answer (if any) is right here? 470 kj or 117 kj? Thank you!

Screenshot 2025-04-12 203210.png


Screenshot 2025-04-12 203309.png
 
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Did you try doing the integral in red?
 
Thread 'Chain falling out of a horizontal tube onto a table'

Thread 'Chain falling out of a horizontal tube onto a table'

My attempt: Initial total M.E = PE of hanging part + PE of part of chain in the tube. I've considered the table as to be at zero of PE. PE of hanging part = ##\frac{1}{2} \frac{m}{l}gh^{2}##. PE of part in the tube = ##\frac{m}{l}(l - h)gh##. Final ME = ##\frac{1}{2}\frac{m}{l}gh^{2}## + ##\frac{1}{2}\frac{m}{l}hv^{2}##. Since Initial ME = Final ME. Therefore, ##\frac{1}{2}\frac{m}{l}hv^{2}## = ##\frac{m}{l}(l-h)gh##. Solving this gives: ## v = \sqrt{2g(l-h)}##. But the answer in the book...
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