Initial program 13.3
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
Initial simplification13.2
\[\leadsto \sqrt{(\left(\frac{\frac{M}{2}}{\frac{d}{D}} \cdot \frac{\frac{M}{2}}{\frac{d}{D}}\right) \cdot \left(-\frac{h}{\ell}\right) + 1)_*} \cdot w0\]
- Using strategy
rm Applied div-inv13.3
\[\leadsto \sqrt{(\left(\color{blue}{\left(\frac{M}{2} \cdot \frac{1}{\frac{d}{D}}\right)} \cdot \frac{\frac{M}{2}}{\frac{d}{D}}\right) \cdot \left(-\frac{h}{\ell}\right) + 1)_*} \cdot w0\]
- Using strategy
rm Applied add-sqr-sqrt13.3
\[\leadsto \sqrt{\color{blue}{\sqrt{(\left(\left(\frac{M}{2} \cdot \frac{1}{\frac{d}{D}}\right) \cdot \frac{\frac{M}{2}}{\frac{d}{D}}\right) \cdot \left(-\frac{h}{\ell}\right) + 1)_*} \cdot \sqrt{(\left(\left(\frac{M}{2} \cdot \frac{1}{\frac{d}{D}}\right) \cdot \frac{\frac{M}{2}}{\frac{d}{D}}\right) \cdot \left(-\frac{h}{\ell}\right) + 1)_*}}} \cdot w0\]
Applied rem-sqrt-square13.3
\[\leadsto \color{blue}{\left|\sqrt{(\left(\left(\frac{M}{2} \cdot \frac{1}{\frac{d}{D}}\right) \cdot \frac{\frac{M}{2}}{\frac{d}{D}}\right) \cdot \left(-\frac{h}{\ell}\right) + 1)_*}\right|} \cdot w0\]
Simplified11.8
\[\leadsto \left|\color{blue}{\sqrt{(\left(\frac{D \cdot M}{2 \cdot d}\right) \cdot \left(\left(-\frac{h}{\ell}\right) \cdot \frac{D \cdot M}{2 \cdot d}\right) + 1)_*}}\right| \cdot w0\]
- Using strategy
rm Applied div-inv11.8
\[\leadsto \left|\sqrt{(\left(\frac{D \cdot M}{2 \cdot d}\right) \cdot \left(\left(-\color{blue}{h \cdot \frac{1}{\ell}}\right) \cdot \frac{D \cdot M}{2 \cdot d}\right) + 1)_*}\right| \cdot w0\]
Applied distribute-lft-neg-in11.8
\[\leadsto \left|\sqrt{(\left(\frac{D \cdot M}{2 \cdot d}\right) \cdot \left(\color{blue}{\left(\left(-h\right) \cdot \frac{1}{\ell}\right)} \cdot \frac{D \cdot M}{2 \cdot d}\right) + 1)_*}\right| \cdot w0\]
Applied associate-*l*7.9
\[\leadsto \left|\sqrt{(\left(\frac{D \cdot M}{2 \cdot d}\right) \cdot \color{blue}{\left(\left(-h\right) \cdot \left(\frac{1}{\ell} \cdot \frac{D \cdot M}{2 \cdot d}\right)\right)} + 1)_*}\right| \cdot w0\]
- Using strategy
rm Applied add-cube-cbrt7.9
\[\leadsto \left|\sqrt{(\left(\frac{D \cdot M}{2 \cdot d}\right) \cdot \left(\left(-h\right) \cdot \left(\frac{1}{\ell} \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{D \cdot M}{2 \cdot d}} \cdot \sqrt[3]{\frac{D \cdot M}{2 \cdot d}}\right) \cdot \sqrt[3]{\frac{D \cdot M}{2 \cdot d}}\right)}\right)\right) + 1)_*}\right| \cdot w0\]
Applied associate-*r*7.9
\[\leadsto \left|\sqrt{(\left(\frac{D \cdot M}{2 \cdot d}\right) \cdot \left(\left(-h\right) \cdot \color{blue}{\left(\left(\frac{1}{\ell} \cdot \left(\sqrt[3]{\frac{D \cdot M}{2 \cdot d}} \cdot \sqrt[3]{\frac{D \cdot M}{2 \cdot d}}\right)\right) \cdot \sqrt[3]{\frac{D \cdot M}{2 \cdot d}}\right)}\right) + 1)_*}\right| \cdot w0\]
Final simplification7.9
\[\leadsto \left|\sqrt{(\left(\frac{D \cdot M}{2 \cdot d}\right) \cdot \left(\left(\left(\frac{-1}{\ell} \cdot \left(\sqrt[3]{\frac{D \cdot M}{2 \cdot d}} \cdot \sqrt[3]{\frac{D \cdot M}{2 \cdot d}}\right)\right) \cdot \sqrt[3]{\frac{D \cdot M}{2 \cdot d}}\right) \cdot h\right) + 1)_*}\right| \cdot w0\]
