A cylindrical reservoir of diameter 4ft ang height 6ft is half-full of water weighing 10lb/ft³. Find the work done emptying the water over the top

Mark Anthony Ramilo
2022-07-14

A cylindrical reservoir of diameter 4ft ang height 6ft is half-full of water weighing 10lb/ft³. Find the work done emptying the water over the top

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$C=f\left(x\right)\frac{{y}^{\prime}}{\sqrt{1+{y}^{\prime 2}}}$

Where f(x) is some function in x and C is a constant.

I am having a miserable time. I tried separation of variables but I end up getting a complex expression and that makes no sense:

$\begin{array}{rl}& C\sqrt{1+{y}^{\text{'}2}}=f\left(x\right){y}^{\text{'}}\\ \phantom{\rule{0ex}{0ex}}\u27fa\phantom{\rule{0ex}{0ex}}& {C}^{2}(1+{y}^{\text{'}2})={f}^{2}\left(x\right){y}^{\text{'}2}\\ \phantom{\rule{0ex}{0ex}}\u27fa\phantom{\rule{0ex}{0ex}}& B+B{y}^{\text{'}2}={f}^{2}\left(x\right){y}^{\text{'}2}\\ \phantom{\rule{0ex}{0ex}}\u27fa\phantom{\rule{0ex}{0ex}}& B+B{y}^{\text{'}2}-{f}^{2}\left(x\right){y}^{\text{'}2}=0\\ \phantom{\rule{0ex}{0ex}}\u27fa\phantom{\rule{0ex}{0ex}}& {y}^{\text{'}2}(B-{f}^{2}(x\left)\right)=-B\\ \phantom{\rule{0ex}{0ex}}\u27fa\phantom{\rule{0ex}{0ex}}& {y}^{\text{'}2}=\frac{-B}{B-{f}^{2}\left(x\right)}\\ \phantom{\rule{0ex}{0ex}}\u27fa\phantom{\rule{0ex}{0ex}}& {y}^{\text{'}}=\sqrt{\frac{-B}{B-{f}^{2}\left(x\right)}}\\ \phantom{\rule{0ex}{0ex}}\u27fa\phantom{\rule{0ex}{0ex}}& dy=\sqrt{\frac{-B}{B-{f}^{2}\left(x\right)}}dx\end{array}$

With $B={C}^{2}$.

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