III gimnazjum

Matematyka 2001

P=πr2+πrl   V=31​πr2⋅h   Dodatkowo w każdym stożku tworząca, promień i wysokość tworzą trójkąt prostokątny.     <img src="https://prod.odrabiamy-assets.pl/uploads/content_image/image/30504/3xiVTh4Lpc2WynlGzBrY9w%3D%3D_48fa2d3776707dd19ecf46d03151116a1074ca590e62245573261e14a56648f7.gif" width="286" height="264"> r2+h2=l2       a)   92+402=l2   l2=81+1600   l=1681<svg xmlns="http://www.w3.org/2000/svg" width='400em' height='1.08em' viewBox='0 0 400000 1080' preserveAspectRatio='xMinYMin slice'><path d='M95,702
c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14
c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54
c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10
s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429
c69,-144,104.5,-217.7,106.5,-221
l0 -0
c5.3,-9.3,12,-14,20,-14
H400000v40H845.2724
s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7
c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z
M834 80h400000v40h-400000z'/></svg>​=41 cm

Rozwiązanie 1

OBRÓT WOKÓŁ KRÓTSZEJ PRZYPROSTOKĄTNEJ Otrzymujemy stożek o wysokości 15 cm i promieniu podstawy 9 cm.  V1​=311​⋅π⋅92⋅155=

<img src="https://prod.odrabiamy-assets.pl/uploads/content_image/image/30505/sjCcbv7zllaS4cOdNn8P5g%3D%3D_727dd138977fe03adc41fa280d1c489b35e871d6d1f58fc7918fd8e0e772aff2.jpg" width="400" height="463"> Obliczmy najpierw (z tw. Pitagorasa) długość wysokości trójkąta, która zarazem jest wysokością stożka.  72+h2=252   h2=252−72   h2=(25−7)(25+7)

P=4πR2   V=34​πR3     a) P=4⋅π⋅42=64π cm2      V=34​⋅π⋅43=   3256​π cm3

P=4πR2   V=34​πR3     a)   4πR2=100π   ∣:4π   R2=25   R=25<svg xmlns="http://www.w3.org/2000/svg" width='400em' height='1.08em' viewBox='0 0 400000 1080' preserveAspectRatio='xMinYMin slice'><path d='M95,702
c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14
c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54
c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10
s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429
c69,-144,104.5,-217.7,106.5,-221
l0 -0
c5.3,-9.3,12,-14,20,-14
H400000v40H845.2724
s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7
c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z
M834 80h400000v40h-400000z'/></svg>​=5 cm

P=4πR2   V=34​πR3     a)   34​πR3=36π   ∣:π   34​R3=36   ∣⋅43​   R3=369⋅413​   R3=27   R=327<svg xmlns="http://www.w3.org/2000/svg" width='400em' height='1.08em' viewBox='0 0 400000 1080' preserveAspectRatio='xMinYMin slice'><path d='M95,702
c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14
c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54
c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10
s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429
c69,-144,104.5,-217.7,106.5,-221
l0 -0
c5.3,-9.3,12,-14,20,-14
H400000v40H845.2724
s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7
c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z
M834 80h400000v40h-400000z'/></svg>​=3 cm

Kula o maksymalnej objetości to kula o promieniu 3 cm (większa się nie zmieści).  Vwalca​=π⋅32⋅6=   π⋅9⋅6=54π cm3

Obliczany kolejno objętości stożka, walca i kuli: Vs​=311​⋅π⋅62⋅124=