$\mathrm{Optimization}:-\mathrm{Minimize}\left(x^2\+\sin \left(x\+y\right)\,\left\{x\+2 y\=9\right\}\right)$

$\left[\mathrm{-0.980004457937920792}\,\left[x=0.0469482872699630\,y=4.47652585636502\right]\right]$

weight of a mechanical device

cost of production

or energy required for a process

dimensions of a mechanical device, or the allowable stresses

minimum and maximum process temperatures

or amount of base materials

$$\begin{gathered} \mathrm{restart}\: \\ \mathrm{with}\left(\mathrm{Optimization}\right)\: \end{gathered}$$

$\mathrm{obj}≔\frac{1}{2}\cdot 4\cdot \mathrm{\pi }\cdot {\mathrm{R}}^2\+2\cdot \mathrm{\pi }\cdot \mathrm{R}\cdot \mathrm{L}\+\mathrm{\pi }\cdot \mathrm{R}\cdot \sqrt{{\mathrm{H}}^2\+{\mathrm{R}}^2}\:$

$\mathrm{cons1}≔\frac{1}{2}\frac{4}{3}\mathrm{\pi }\cdot {\mathrm{R}}^3\+\mathrm{\pi } {\mathrm{R}}^2\cdot \mathrm{L}\+\frac{1}{3}\cdot \mathrm{\pi }\cdot {\mathrm{R}}^2\cdot \mathrm{H}\=3⟦{\mathrm{m}}^3⟧\:$

$\mathrm{cons2}≔0\le \mathrm{R}\,0\le \mathrm{L}\,0\le \mathrm{H}\:$

$\mathrm{dimensions}≔\mathrm{Minimize}\left(\mathrm{obj}\,\left\{\mathrm{cons1}\,\mathrm{cons2}\right\}\,\mathrm{initialpoint}\=\left\{\mathrm{H}\=1⟦\mathrm{m}⟧\,\mathrm{L}\=1⟦\mathrm{m}⟧\,\mathrm{R}\=1⟦\mathrm{m}⟧\right\}\right)$

$\left[10.2533536615869920⟦{\mathrm{m}}^2⟧\,\left[\mathrm{H}\=0.785093823049978⟦\mathrm{m}⟧\,\mathrm{L}\=0.392546902492684⟦\mathrm{m}⟧\,\mathrm{R}\=0.877761593519080⟦\mathrm{m}⟧\right]\right]$

$\mathrm{eval}\left(\left[\mathrm{cons1}\right]\,\mathrm{dimensions}\left[2\right]\right)$

$\left[3.00000000039170{⟦\mathrm{m}⟧}^3\=3⟦{\mathrm{m}}^3⟧\right]$