III gimnazjum

Matematyka 2001

Matematyka

n &nbsp;-liczba naturalna n,n+1,n+2 &nbsp;-trzy kolejne liczby naturalne 4n &nbsp;- pierwsza liczba w kodzie do komputera Jacka 4n+1

Wiemy, że cena książki z 5-procentowym podatkiem VAT jest równa 21 zł zatem: x &nbsp;- cena książki bez podatku 1,05x=21  ∣⋅100

Rysunek pomocniczy: <img class="img-responsive img-thumbnail dropzone-img" src="https://prod.odrabiamy-assets.pl/uploads/content_image/image/171207/snet4Ot1fRJFtNoeXVc0pw%3D%3D_2a1ee5384adc43305cce351afd7bc02963488bceb7e345877e53f11041191b1a.jpg" width="358" height="352"> Obliczmy długość odcinak oznaczonego literą&nbsp; x &nbsp;: 32+x2=52 &nbsp; 9+x2=25  ∣−9 &nbsp; x2=16  ∣<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>​ &nbsp; x=4 &nbsp; Obliczmy pole powierzchni całkowitej tej bryły: (czyli sumę pól bocznych dwóch stożków i jednego walca) Pc​=2⋅π⋅4⋅5+2π⋅4⋅4=40π+32π=72π &nbsp;

Podstawa tego ostrosłupa jest trójkątem prostokątnym i równoramiennym zatem: Rysunek pomocniczy: <img class="img-responsive img-thumbnail dropzone-img" src="https://prod.odrabiamy-assets.pl/uploads/content_image/image/171208/d/ZjAdfCREPI1pRSRyEiFg%3D%3D_2824a2d77164ff9fd56a193e92635816a5c29dfb71ebce6cd375593fb8f88aad.jpg" width="328" height="289"> Korzystając&nbsp; z twierdzenia Pitagorasa możemy obliczyć długość krawędzi oznaczonej literą&nbsp; b &nbsp; 52+102=b2 &nbsp; 25+100=b2 &nbsp; b2=125  ∣<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>​ &nbsp; b=125<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>​ &nbsp; b=55<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>​ &nbsp;

Korzystamy z wzoru: s=v⋅t &nbsp; Wiemy, że samolot pokonuje pewną trasę w czasie&nbsp; 3 &nbsp;h i&nbsp; 40 &nbsp;minut. v &nbsp;-prędkość samolotu, z jaką pokona on określoną długość trasy w czasie&nbsp; 3

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