Question

Zadanie optymalizacyjne z matematyki. Proszę o rozwiązanie i wyjaśnienie

Answer 1

Volume of the pyramid:

`V=\frac{s^2h}3`

Perimeter of the cross-section:

`40=\sqrt2\,s+2\sqrt{\frac{s^2}2+h^2}=\sqrt2\left(s+√(s^2+2h^2)\right)`

`\implies h=\sqrt{\frac{(20\sqrt2-s)^2-s^2}2}=√(400-20\sqrt2\,s)`

Area of the cross-section:

`P=\frac12(\sqrt2\,s)h=(sh)/(\sqrt2)`

`\implies P=(s√(400-20\sqrt2\,s))/(\sqrt2)=s√(200-10\sqrt2\,s)`

First derivative test:

`(\mathrm dP)/(\mathrm ds)=\frac{20\sqrt2-3s}{\sqrt{4-\frac{\sqrt2}5s}}=0\implies s=\frac{20\sqrt2}3`

Then the height of the cross-section/pyramid is

`h=√(400-20\sqrt2\,s)=(20)/(\sqrt3)`

The volume of the pyramid that maximizes the cross-sectional area
`P` is

`V=\frac{\left(\frac{20\sqrt2}3\right)^2(20)/(\sqrt3)}3=(16000)/(27\sqrt3)`