zadanie na ferie :P :P :P
prosze o rozwiazanie dwoch zadan daje dwa zalaczniki na jednym prosze zadanie 5 a na drugim 3 plssss o wyjasnienie DAJE NAJ
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Zad. 5 załącznik pierwszy
Długość jednej wskazówki = 4cm
Długość drugiej wskazówki = 3 cm
Aby wyliczyć drogę, jaką pokonują wskazówki w ciągu 12 godzin wystarczy skorzystać ze wzoru na obwód koła:
Nasz promień "r" to nic innego, jak długości wskazówek
Pierwsza wskazówka pokonuje długość =![2*\pi*4cm=8*\pi 2*\pi*4cm=8*\pi](https://tex.z-dn.net/?f=2%2A%5Cpi%2A4cm%3D8%2A%5Cpi)
gdyż:![\pi \approx 3,14cm \pi \approx 3,14cm](https://tex.z-dn.net/?f=%5Cpi+%5Capprox+3%2C14cm)
Druga wskazówka pokonuje długość =![2*\pi*3cm=6*\pi 2*\pi*3cm=6*\pi](https://tex.z-dn.net/?f=2%2A%5Cpi%2A3cm%3D6%2A%5Cpi)
zad. 3
Zarówno zad. 5, jak i to jest na wykorzystanie wzoru na obwód koła.
Obwód koła =![2*\pi * r 2*\pi * r](https://tex.z-dn.net/?f=2%2A%5Cpi+%2A+r)
a) średnica koła to nic innego jak dwa razy promień koła.
*) Śrenica koła =![\frac{3}{4} \frac{3}{4}](https://tex.z-dn.net/?f=%5Cfrac%7B3%7D%7B4%7D+)
Promień koła "r" =![\frac{1}{2} * \frac{3}{4} \frac{1}{2} * \frac{3}{4}](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B2%7D+%2A+%5Cfrac%7B3%7D%7B4%7D)
Obwód koła =![2*\pi * \frac{3}{8} = 0,75\pi j. 2*\pi * \frac{3}{8} = 0,75\pi j.](https://tex.z-dn.net/?f=2%2A%5Cpi+%2A+%5Cfrac%7B3%7D%7B8%7D+%3D+0%2C75%5Cpi+j.)
*) Promień koła "r" =![\frac{1}{2} * 6 = 3 \frac{1}{2} * 6 = 3](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B2%7D+%2A+6+%3D+3)
Obwód koła =![2*\pi * 3 = 6\pi j. 2*\pi * 3 = 6\pi j.](https://tex.z-dn.net/?f=2%2A%5Cpi+%2A+3+%3D+6%5Cpi+j.)
*) Promień koła "r" =![\frac{1}{2} * 123 = 61,5 \frac{1}{2} * 123 = 61,5](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B2%7D+%2A+123+%3D+61%2C5)
Obwód koła =![2*\pi * 61,5 = 123\pi j. 2*\pi * 61,5 = 123\pi j.](https://tex.z-dn.net/?f=2%2A%5Cpi+%2A+61%2C5+%3D+123%5Cpi+j.)
*) Promień koła "r" =![\frac{1}{2} * 4,2=2,1 \frac{1}{2} * 4,2=2,1](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B2%7D+%2A+4%2C2%3D2%2C1)
Obwód koła =![2*\pi * 2,1 = 4,2\pi j. 2*\pi * 2,1 = 4,2\pi j.](https://tex.z-dn.net/?f=2%2A%5Cpi+%2A+2%2C1+%3D+4%2C2%5Cpi+j.)
*) Promień koła "r" =![\frac{1}{2} * 3\pi = \frac{3}{2} \pi \frac{1}{2} * 3\pi = \frac{3}{2} \pi](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B2%7D+%2A+3%5Cpi+%3D+%5Cfrac%7B3%7D%7B2%7D+%5Cpi)
Obwód koła =![2*\pi * \frac{3}{2} \pi = 3\pi^{2} j. 2*\pi * \frac{3}{2} \pi = 3\pi^{2} j.](https://tex.z-dn.net/?f=2%2A%5Cpi+%2A+%5Cfrac%7B3%7D%7B2%7D+%5Cpi+%3D+3%5Cpi%5E%7B2%7D+j.)
*) Promień koła "r" =![\frac{1}{2} * 3\sqrt{2} = \frac{3}{2} \sqrt{2} \frac{1}{2} * 3\sqrt{2} = \frac{3}{2} \sqrt{2}](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B2%7D+%2A+3%5Csqrt%7B2%7D+%3D+%5Cfrac%7B3%7D%7B2%7D+%5Csqrt%7B2%7D)
Obwód koła =![2*\pi * \frac{3}{2} \sqrt{2} = 3\sqrt{2} j. 2*\pi * \frac{3}{2} \sqrt{2} = 3\sqrt{2} j.](https://tex.z-dn.net/?f=2%2A%5Cpi+%2A+%5Cfrac%7B3%7D%7B2%7D+%5Csqrt%7B2%7D+%3D+3%5Csqrt%7B2%7D+j.)
*) Promień koła "r" =![\frac{1}{2} *\frac{\sqrt{5} }{2} = \frac{\sqrt{5} }{4} \frac{1}{2} *\frac{\sqrt{5} }{2} = \frac{\sqrt{5} }{4}](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B2%7D+%2A%5Cfrac%7B%5Csqrt%7B5%7D+%7D%7B2%7D+%3D+%5Cfrac%7B%5Csqrt%7B5%7D+%7D%7B4%7D)
Obwód koła =![2*\pi * \frac{3}{2} \sqrt{2} = 3\sqrt{2} j. 2*\pi * \frac{3}{2} \sqrt{2} = 3\sqrt{2} j.](https://tex.z-dn.net/?f=2%2A%5Cpi+%2A+%5Cfrac%7B3%7D%7B2%7D+%5Csqrt%7B2%7D+%3D+3%5Csqrt%7B2%7D+j.)
b)![r = \frac{1}{2} r = \frac{1}{2}](https://tex.z-dn.net/?f=r+%3D+%5Cfrac%7B1%7D%7B2%7D)
Obwód koła \approx![2*3,14 * \frac{1}{2} \approx 3,14 j. 2*3,14 * \frac{1}{2} \approx 3,14 j.](https://tex.z-dn.net/?f=2%2A3%2C14+%2A+%5Cfrac%7B1%7D%7B2%7D+%5Capprox+3%2C14+j.)
gdzie: j - jakaś jednostka (może to być cm, m, km lub jakaś inna jednostka miary)
*) Obwód koła \approx![2*3,14 * \frac{5}{4} \approx 7,85 j. 2*3,14 * \frac{5}{4} \approx 7,85 j.](https://tex.z-dn.net/?f=2%2A3%2C14+%2A+%5Cfrac%7B5%7D%7B4%7D+%5Capprox+7%2C85+j.)
*) Obwód koła \approx![2*3,14 * 3 \approx 18,84 j. 2*3,14 * 3 \approx 18,84 j.](https://tex.z-dn.net/?f=2%2A3%2C14+%2A+3+%5Capprox+18%2C84+j.)
*) Obwód koła \approx![2*3,14 * 111 \approx 687,08 j. 2*3,14 * 111 \approx 687,08 j.](https://tex.z-dn.net/?f=2%2A3%2C14+%2A+111+%5Capprox+687%2C08+j.)
*) Obwód koła \approx![2*3,14 * \frac{5}{4} \approx 7,85 j. 2*3,14 * \frac{5}{4} \approx 7,85 j.](https://tex.z-dn.net/?f=2%2A3%2C14+%2A+%5Cfrac%7B5%7D%7B4%7D+%5Capprox+7%2C85+j.)
*) Obwód koła =![2*\pi * \pi = 2\pi^{2} \approx 2* (3,14)^{2} \approx 19,7192 j. 2*\pi * \pi = 2\pi^{2} \approx 2* (3,14)^{2} \approx 19,7192 j.](https://tex.z-dn.net/?f=2%2A%5Cpi+%2A+%5Cpi+%3D+2%5Cpi%5E%7B2%7D+%5Capprox+2%2A+%283%2C14%29%5E%7B2%7D+%5Capprox+19%2C7192+j.)
*) Obwód koła =![2*\pi * \sqrt{2} \approx 2*3,14 * \sqrt{2} \approx 6,28\sqrt{2} j. 2*\pi * \sqrt{2} \approx 2*3,14 * \sqrt{2} \approx 6,28\sqrt{2} j.](https://tex.z-dn.net/?f=2%2A%5Cpi+%2A+%5Csqrt%7B2%7D+%5Capprox+2%2A3%2C14+%2A+%5Csqrt%7B2%7D+%5Capprox+6%2C28%5Csqrt%7B2%7D+j.)
*) Obwód koła =![2*\pi * \frac{\sqrt{3}}{2} \approx 2*3,14 * \frac{\sqrt{3}}{2} \approx 3,14\sqrt{3} j. 2*\pi * \frac{\sqrt{3}}{2} \approx 2*3,14 * \frac{\sqrt{3}}{2} \approx 3,14\sqrt{3} j.](https://tex.z-dn.net/?f=2%2A%5Cpi+%2A+%5Cfrac%7B%5Csqrt%7B3%7D%7D%7B2%7D+%5Capprox+2%2A3%2C14+%2A+%5Cfrac%7B%5Csqrt%7B3%7D%7D%7B2%7D+%5Capprox+3%2C14%5Csqrt%7B3%7D+j.)
*) Obwód koła =![2*\pi * (1+\sqrt{5}) \approx6,28 * (1+\sqrt{5}) \approx 6,28 + 6,28\sqrt{5} j. 2*\pi * (1+\sqrt{5}) \approx6,28 * (1+\sqrt{5}) \approx 6,28 + 6,28\sqrt{5} j.](https://tex.z-dn.net/?f=2%2A%5Cpi+%2A+%281%2B%5Csqrt%7B5%7D%29+%5Capprox6%2C28+%2A+%281%2B%5Csqrt%7B5%7D%29+%5Capprox+6%2C28+%2B+6%2C28%5Csqrt%7B5%7D+j.)
c) podaj promień koła o obwodzie:
*) Obwód koła [Obw.]= 8\pi Obw. =![2*\pi * r 2*\pi * r](https://tex.z-dn.net/?f=2%2A%5Cpi+%2A+r)
![r = \frac{Obw.}{2*\pi} j. r = \frac{Obw.}{2*\pi} j.](https://tex.z-dn.net/?f=r+%3D+%5Cfrac%7BObw.%7D%7B2%2A%5Cpi%7D+j.)
*)![r = \frac{1,2\pi}{2*\pi} = 0,6\pi j. r = \frac{1,2\pi}{2*\pi} = 0,6\pi j.](https://tex.z-dn.net/?f=r+%3D+%5Cfrac%7B1%2C2%5Cpi%7D%7B2%2A%5Cpi%7D+%3D+0%2C6%5Cpi+j.)
*)![r = \frac{\frac{5}{6}\pi}{2*\pi} = \frac{5}{12}\pi j. r = \frac{\frac{5}{6}\pi}{2*\pi} = \frac{5}{12}\pi j.](https://tex.z-dn.net/?f=r+%3D+%5Cfrac%7B%5Cfrac%7B5%7D%7B6%7D%5Cpi%7D%7B2%2A%5Cpi%7D+%3D+%5Cfrac%7B5%7D%7B12%7D%5Cpi+j.)
*)![r = \frac{\frac{1}{2}}{2*\pi} = \pi j. r = \frac{\frac{1}{2}}{2*\pi} = \pi j.](https://tex.z-dn.net/?f=r+%3D+%5Cfrac%7B%5Cfrac%7B1%7D%7B2%7D%7D%7B2%2A%5Cpi%7D+%3D+%5Cpi+j.)
*)![r = \frac{4\sqrt{3}}{2*\pi} = 2\sqrt{3}\pi j. r = \frac{4\sqrt{3}}{2*\pi} = 2\sqrt{3}\pi j.](https://tex.z-dn.net/?f=r+%3D+%5Cfrac%7B4%5Csqrt%7B3%7D%7D%7B2%2A%5Cpi%7D+%3D+2%5Csqrt%7B3%7D%5Cpi+j.)
d) Podaj średnicę koła o obwodzie...
Średnica koła [Sr.] =![\frac{1}{2} *r \frac{1}{2} *r](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B2%7D+%2Ar)
Obwód koła [Obw.] =![4*\pi*Sr. 4*\pi*Sr.](https://tex.z-dn.net/?f=4%2A%5Cpi%2ASr.)
*) Średnica koła![Sr. = \frac{3\pi}{4*\pi} = \frac{3}{4}j. Sr. = \frac{3\pi}{4*\pi} = \frac{3}{4}j.](https://tex.z-dn.net/?f=Sr.+%3D+%5Cfrac%7B3%5Cpi%7D%7B4%2A%5Cpi%7D+%3D+%5Cfrac%7B3%7D%7B4%7Dj.)
*) Średnica koła![Sr. = \frac{8,6\pi}{4*\pi} = 2,15j. Sr. = \frac{8,6\pi}{4*\pi} = 2,15j.](https://tex.z-dn.net/?f=Sr.+%3D+%5Cfrac%7B8%2C6%5Cpi%7D%7B4%2A%5Cpi%7D+%3D+2%2C15j.)
*) Średnica koła![Sr. = \frac{\frac{4}{11}\pi}{4*\pi} = \frac{1}{11}j. Sr. = \frac{\frac{4}{11}\pi}{4*\pi} = \frac{1}{11}j.](https://tex.z-dn.net/?f=Sr.+%3D+%5Cfrac%7B%5Cfrac%7B4%7D%7B11%7D%5Cpi%7D%7B4%2A%5Cpi%7D+%3D+%5Cfrac%7B1%7D%7B11%7Dj.)
*) Średnica koła![Sr. = \frac{\frac{2}{5}}{4*\pi} = \frac{1}{10\pi}j. Sr. = \frac{\frac{2}{5}}{4*\pi} = \frac{1}{10\pi}j.](https://tex.z-dn.net/?f=Sr.+%3D+%5Cfrac%7B%5Cfrac%7B2%7D%7B5%7D%7D%7B4%2A%5Cpi%7D+%3D+%5Cfrac%7B1%7D%7B10%5Cpi%7Dj.)
*) Średnica koła![Sr. = \frac{5\sqrt{2}}{4*\pi}j. Sr. = \frac{5\sqrt{2}}{4*\pi}j.](https://tex.z-dn.net/?f=Sr.+%3D+%5Cfrac%7B5%5Csqrt%7B2%7D%7D%7B4%2A%5Cpi%7Dj.)