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Pe: 2011-05-01T16:50:51+03:00 In: In:

geometrie

1.Aflati raportul dintre volumul unui corp sferic de raza R si volumul unei prisme hexagonale drepte ale carei muchii sant congruente cu raza acestui corp sferic.

2.Fermierii au depozitat graul recoltat in doua gramezi de forma conica , ale caror inaltimi si diametre se raporta 1:2. Aflati raportul dintre volumele respective ale acestor gramezi de grau.
Multumesc.

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2 raspunsuri

  1. sindex

    1)
    { {V_s} = vol.sfera \cr {V_{p.h.}} = vol.prisma \cr \left. \matrix{ {V_s} = {{4\pi {R^3}} \over 3} \hfill \cr {V_{p.h.}} = 6{{{R^2}\sqrt 3 } \over 4} \cdot R = {{3{R^3}\sqrt 3 } \over 2} \hfill \cr} \right| \Rightarrow {{{V_s}} \over {{V_{p.h.}}}} = {{4\pi {R^3}} \over 3} \cdot {2 \over {3{R^3}\sqrt 3 }} = {{8\pi } \over {9\sqrt 3 }} \cr}

  2. sindex

    2)

    { {{{d_1}} \over {{d_2}}} = {1 \over 2} \Leftrightarrow {{2{r_1}} \over {2{r_2}}} = {1 \over 2} \Rightarrow {{{r_1}} \over {{r_2}}} = {1 \over 2} \cr \left. \matrix{ {V_1} = {{\pi {r_1}^2{h_1}} \over 3} \hfill \cr {V_2} = {{\pi {r_2}^2{h_2}} \over 3} \hfill \cr} \right| \Rightarrow {{{V_1}} \over {{V_2}}} = {{{r_1}^2{h_1}} \over {{r_2}^2{h_2}}} = {\left( {{{{r_1}} \over {{r_2}}}} \right)^2} \cdot {{{h_1}} \over {{h_2}}} = {\left( {{1 \over 2}} \right)^2} \cdot {1 \over 2} = {1 \over 8} \cr}

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