III gimnazjum

Matematyka wokół nas 3

24cm=2Πr1​ r1​=π12​cm 12cm=2Πr1​ r1​=Π6​cm Odp. Składając kartkę wzdłuż dłuższego boku promień podstawy będzie miał Π12​ cm, natomiast wzdłuż krótszego Π6​ cm.

Rozwiązanie 1

<img src="https://prod.odrabiamy-assets.pl/uploads/content_image/image/24169/KnUWYSDKGnfPnb0oN5O%2BDA%3D%3D_1e45524e0dfb345187e6694e6bf3cb55ca65106251198c38c661ffa3c477d66b.jpg" width="339" height="320"> d=122<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>​cm 2a2=(122<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>​)2

<img src="https://prod.odrabiamy-assets.pl/uploads/content_image/image/24171/6mn2/lQ45iXE/CSRJX/PBg%3D%3D_dbf23eae4dd471df1a396bdcb15eb3a2d9b1a28706011ec99e4f2fde324c8655.jpg" width="414" height="480"> ∣BD∣=123<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>​cm ∣∠ABD∣=60o ∣AB∣=21​⋅123<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>​=63<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>​cm

<img src="https://prod.odrabiamy-assets.pl/uploads/content_image/image/24172/8SeGItvg9qtBr8KHpIlDKw%3D%3D_010d7e5a2849da7d51853b4fac53544ddb199438b2681b8eb0bde4f3f0055cda.jpg" width="399" height="442"> 2r=12cm b=15cm r=6cm a=2⋅6Π=12Πcm≈37,68

<img src="https://prod.odrabiamy-assets.pl/uploads/content_image/image/24175/ChjorUOUv0%2BjjgJF4ijLlg%3D%3D_1c888e9e8ac61b7f9b7276ab346f019b78891d6d9c4f31e0df06b38a1695b26a.jpg" width="388" height="419"> a=2r 2Πr=6Π

I przypadek <img src="https://prod.odrabiamy-assets.pl/uploads/content_image/image/25090/McEhwlXpMpvRlRa%2B76MLXA%3D%3D_5ea19255bdebd4bc7ac5247afefbc1d23e351a1949ddc109553e578e9f9c3fb5.jpg" width="303" height="167"> Ob=80π cm   a - dłuższy bok prostokąta (długość ta jest także obwodem podstawy walca) b - krótszy bok prostokąta (długość ta jest również wysokością walca) Krótszy bok stanowi 2/3 dłuższego boku. Stąd: b=32​a   Obwód prostokąta obliczamy ze wzoru: Ob=2a+2b   Podstawmy dane do powyższego wzoru: 80π=2a+2⋅32​a   80π=2a+34​a   80π=310​a        ∣⋅103​

<img src="https://prod.odrabiamy-assets.pl/uploads/content_image/image/25102/ELatz%2BQuSBj4sxuFugsVdQ%3D%3D_550940f1f949e00e70ed1a6e7d706814c591d41a0ef84a472cd2aeeaff3b5caf.jpg" width="170" height="480"> 32+x2=52 x2=25−9

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