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a)&nbsp;Przyjmijmy oznaczenia jak na poniższym rysunku. <img src="https://prod.odrabiamy-assets.pl/uploads/content_image/image/126487/Lk73UV7yAyAMgSTKVivtAA%3D%3D_04d38da7ca3b24d105c3436cb9afa2f0c38b7bc283157483e59abbf6a4e149d6.jpg" width="255" height="338">&nbsp;

a) Przyjmijmy oznaczenia jak na poniższym rysunku. <img src="https://prod.odrabiamy-assets.pl/uploads/content_image/image/126530/VW%2BCO/lekb1Z31r%2BTchuRA%3D%3D_5f71b063803411e1bc64451ffa289da0d4f81829dd6548788d18b1f68e4b7004.jpg" alt="" width="666" height="373">&nbsp;

Przyjmijmy oznaczenia jak na poniższym rysunku.&nbsp;

Wykonajmy rysunek pomocniczy. <img src="https://prod.odrabiamy-assets.pl/uploads/content_image/image/126608/hZXBgWCfVfKuVFZM2J9yyw%3D%3D_e41793510fdebad3e18c846e037c57b7a2085d412d68dcabdf0e7df1b118b7da.jpg" width="444" height="327">&nbsp;

Przyjmijmy oznaczenia jak na poniższym rysunku. <img src="https://prod.odrabiamy-assets.pl/uploads/content_image/image/126498/vRTXX0/sW3tnI7xSi2CQpw%3D%3D_ca858ad96d68255fa4043c4930a80800874471575386bf4fc39bc670ed8b50cf.jpg" alt="" width="333" height="250">&nbsp; Wyznaczamy miarę kąta 𝛼, wiedząc, że suma miar kątów wewnętrznych w każdym trójkącie wynosi 180o. <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/354a9679ac5b5fe80f5b7f97db531cbfb3b1beaa389a4a9da1f63ca3034a7ac2_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/354a9679ac5b5fe80f5b7f97db531cbfb3b1beaa389a4a9da1f63ca3034a7ac2_big.svg" type="image/svg+xml"></object> &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/9256402608541f6ddbf3e48fa38115fa2b3624283772200269ef747e013dd607_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/9256402608541f6ddbf3e48fa38115fa2b3624283772200269ef747e013dd607_big.svg" type="image/svg+xml"></object> &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/62b7de1f2d358949058ef0a61d681eb1ab7d3fb38e7b540b210fed25c7da4f01_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/62b7de1f2d358949058ef0a61d681eb1ab7d3fb38e7b540b210fed25c7da4f01_big.svg" type="image/svg+xml"></object> &nbsp; Korzystamy z zależności trygonometrycznych i wyznaczamy długość promienia podstawy walca. <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/c3b6b80fae8f9dccf256db65c76ee1399b530272c6a76b845cbd78ec884f2a18_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/c3b6b80fae8f9dccf256db65c76ee1399b530272c6a76b845cbd78ec884f2a18_big.svg" type="image/svg+xml"></object> &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/d8e29bc811af6141a2cc44aabe22b87afce08c8c94b6767accd525c724d0b4fb_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/d8e29bc811af6141a2cc44aabe22b87afce08c8c94b6767accd525c724d0b4fb_big.svg" type="image/svg+xml"></object> &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/b74f4bb855faf585f8ce7159a7a10f4286ae4570c4bd9699b9dac560ec8edef4_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/b74f4bb855faf585f8ce7159a7a10f4286ae4570c4bd9699b9dac560ec8edef4_big.svg" type="image/svg+xml"></object> &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/6be16d4358ab1eebbac36e473557f6309147492c9795e11bac0600e16b4e8ebe_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/6be16d4358ab1eebbac36e473557f6309147492c9795e11bac0600e16b4e8ebe_big.svg" type="image/svg+xml"></object> &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/b2a60a64462848f6de0266ad69fba2167d4eed988862c7cdb33ddadee3bc0101_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/b2a60a64462848f6de0266ad69fba2167d4eed988862c7cdb33ddadee3bc0101_big.svg" type="image/svg+xml"></object> &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/83e9d110aaf2f3791834e76fe451d10c4c8f40a09e88b54de935317033126299_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/83e9d110aaf2f3791834e76fe451d10c4c8f40a09e88b54de935317033126299_big.svg" type="image/svg+xml"></object> &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/9e77adfd22e74552d260ffbef3d53c5409aa2996f6b9f1904f25150ef158db62_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/9e77adfd22e74552d260ffbef3d53c5409aa2996f6b9f1904f25150ef158db62_big.svg" type="image/svg+xml"></object> &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/cdabec257f60134b6f1639d15d749f21c7ae613b78a1fbd2e1e4a40b2001f81c_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/cdabec257f60134b6f1639d15d749f21c7ae613b78a1fbd2e1e4a40b2001f81c_big.svg" type="image/svg+xml"></object> &nbsp; Obliczamy objętość walca. <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/2f9cb769d77bd03da97c79d4c3a54b7bede05a2d366a80c7fd930dd856f2fbe7_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/2f9cb769d77bd03da97c79d4c3a54b7bede05a2d366a80c7fd930dd856f2fbe7_big.svg" type="image/svg+xml"></object> &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/d37446619a5019cc112e3ece9337766f42bc7310b439159a10a69329bd7c4408_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/d37446619a5019cc112e3ece9337766f42bc7310b439159a10a69329bd7c4408_big.svg" type="image/svg+xml"></object> &nbsp; Odp. Objętość walca wynosi&nbsp;375/4𝜋.

Powierzchnia boczna walca po rozwinięciu jest prostokątem, którego jeden bok jest dwa razy dłuższy od drugiego. Do rozważenia będziemy mieć dwa przypadki.&nbsp; Przyjmijmy oznaczenia jak na poniższym rysunku. <img src="https://prod.odrabiamy-assets.pl/uploads/content_image/image/126595/uSUlDv1oS2H9ZwXy5WVE7A%3D%3D_e19572b412908fbb282158dd645a4c254e9ca7c880e8de5fd8c9f7b324b6f471.jpg" width="555" height="344"> <hr> Przypadek I Z twierdzenia Pitagorasa otrzymujemy: <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/5d1a8212677b5a29339211ec8303182e98e51296ea76f6dd948441601ab3ac90_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/5d1a8212677b5a29339211ec8303182e98e51296ea76f6dd948441601ab3ac90_big.svg" type="image/svg+xml"></object> &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/8d18e20be9717f9f349c8a2a3086bf716c7923c856c57ef1a6c7bcae24b03769_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/8d18e20be9717f9f349c8a2a3086bf716c7923c856c57ef1a6c7bcae24b03769_big.svg" type="image/svg+xml"></object> &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/b25912a8ab2d55eef1d65550b46e3c885f6fc39e3f58910b77917a065bade3b2_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/b25912a8ab2d55eef1d65550b46e3c885f6fc39e3f58910b77917a065bade3b2_big.svg" type="image/svg+xml"></object> &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/dc2cd4d0de0fe11a1939f2845ab754246ab91b529dad6fffce29eb538e65198f_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/dc2cd4d0de0fe11a1939f2845ab754246ab91b529dad6fffce29eb538e65198f_big.svg" type="image/svg+xml"></object> &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/50cc010b4c00ca003dfa193d9202c8681dd68c8ddaed53f0593cb30b18cdd94e_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/50cc010b4c00ca003dfa193d9202c8681dd68c8ddaed53f0593cb30b18cdd94e_big.svg" type="image/svg+xml"></object> &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/a5a05036cd034a626700555b4d214b1467084013312c11635ad41a6424339dd2_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/a5a05036cd034a626700555b4d214b1467084013312c11635ad41a6424339dd2_big.svg" type="image/svg+xml"></object> &nbsp; Obwód podstawy walca jest równy a, zatem: <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/b000939fa8549dc38ab70088f3c073604cbdc0baf05c382d08ffa036426211f5_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/b000939fa8549dc38ab70088f3c073604cbdc0baf05c382d08ffa036426211f5_big.svg" type="image/svg+xml"></object> &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/3d7b4bcb3fd11103ee968d5f602431332974bf79cfce0dd279a6dd984442b385_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/3d7b4bcb3fd11103ee968d5f602431332974bf79cfce0dd279a6dd984442b385_big.svg" type="image/svg+xml"></object> &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/ba6ae7d67f31cf3fdc855195ddf40e0227f0409b12df86a5da65e5af8e1b4b59_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/ba6ae7d67f31cf3fdc855195ddf40e0227f0409b12df86a5da65e5af8e1b4b59_big.svg" type="image/svg+xml"></object> &nbsp; Obliczamy pole podstawy walca. <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/41fb59ce2ba3fedff1913d52a207c8e37ce583b302819931e421966db7190181_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/41fb59ce2ba3fedff1913d52a207c8e37ce583b302819931e421966db7190181_big.svg" type="image/svg+xml"></object> &nbsp; Obliczamy pole powierzchni bocznej walca. <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/ad6155e26e3bbda783b91b985f3fcc8d31c712f9681f68cb514660db0fb98a78_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/ad6155e26e3bbda783b91b985f3fcc8d31c712f9681f68cb514660db0fb98a78_big.svg" type="image/svg+xml"></object> &nbsp; Obliczamy pole powierzchni całkowitej walca. <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/29d2db4ee28abc8ce03f9cc4b1251b773f1df21b29d7e4f98cd51fd3f445cc90_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/29d2db4ee28abc8ce03f9cc4b1251b773f1df21b29d7e4f98cd51fd3f445cc90_big.svg" type="image/svg+xml"></object> &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/31feebef18a8d559efe65786efb9b96c4a99a6f049b8bb2fadca4736ef4ff204_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/31feebef18a8d559efe65786efb9b96c4a99a6f049b8bb2fadca4736ef4ff204_big.svg" type="image/svg+xml"></object> &nbsp; <hr> Przypadek II Z twierdzenia Pitagorasa otrzymujemy: <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/0e2f2875a97a89c0f462d6100f09ee2875f67a65d562cb1020095966eeada18b_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/0e2f2875a97a89c0f462d6100f09ee2875f67a65d562cb1020095966eeada18b_big.svg" type="image/svg+xml"></object> &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/6577205c302cdc836ef01d7cfabf805178869c8c6b667e178b88c09e7aaeba0c_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/6577205c302cdc836ef01d7cfabf805178869c8c6b667e178b88c09e7aaeba0c_big.svg" type="image/svg+xml"></object> &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/7ddfb0403e2737ee47a52f4629b791a777f49993ba818da0ecc432181b0fcf08_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/7ddfb0403e2737ee47a52f4629b791a777f49993ba818da0ecc432181b0fcf08_big.svg" type="image/svg+xml"></object> &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/1b49ff5192b44303a6812149cccf5356b3e531166744a710531f84fdc434426b_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/1b49ff5192b44303a6812149cccf5356b3e531166744a710531f84fdc434426b_big.svg" type="image/svg+xml"></object> &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/58d05db67a350b497f466c9a2292961dd943a86bcce1cb917fdb7122e80b7845_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/58d05db67a350b497f466c9a2292961dd943a86bcce1cb917fdb7122e80b7845_big.svg" type="image/svg+xml"></object> &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/1635099f9ce11cd920dc19e753d7bcbd42fade26568c194ea66048e7448d23fe_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/1635099f9ce11cd920dc19e753d7bcbd42fade26568c194ea66048e7448d23fe_big.svg" type="image/svg+xml"></object> &nbsp; Obwód podstawy walca jest równy 2b, zatem: <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/d78001f24230812ad87c7ff08849eb20cdc46aa1d744b07f89e5adb9a6bc9a20_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/d78001f24230812ad87c7ff08849eb20cdc46aa1d744b07f89e5adb9a6bc9a20_big.svg" type="image/svg+xml"></object> &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/6dc82cd02f91979ceafa756ff46157f9559d418bc4f498f52881cdb6913fddcb_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/6dc82cd02f91979ceafa756ff46157f9559d418bc4f498f52881cdb6913fddcb_big.svg" type="image/svg+xml"></object> &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/a10c4cdee2d2877a1720ab3e10bd298897192e051ded8c79569495aef05f9437_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/a10c4cdee2d2877a1720ab3e10bd298897192e051ded8c79569495aef05f9437_big.svg" type="image/svg+xml"></object> &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/b03088aec2fbe060799962b1771ee0a3988e25853199ddc6a2683d91bb157b70_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/b03088aec2fbe060799962b1771ee0a3988e25853199ddc6a2683d91bb157b70_big.svg" type="image/svg+xml"></object> &nbsp; Obliczamy pole podstawy walca. <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/184a90d7deb37eb239fb2fef7760378eadd8f889081de9df12eb4ace851c84d1_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/184a90d7deb37eb239fb2fef7760378eadd8f889081de9df12eb4ace851c84d1_big.svg" type="image/svg+xml"></object> &nbsp; Obliczamy pole powierzchni bocznej walca. <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/16b6d7029fa486d15fcbf997b45297c58b6b7b6d9b09c45b34585edfbc90b5fa_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/16b6d7029fa486d15fcbf997b45297c58b6b7b6d9b09c45b34585edfbc90b5fa_big.svg" type="image/svg+xml"></object> &nbsp; Obliczamy pole powierzchni całkowitej walca. <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/29d2db4ee28abc8ce03f9cc4b1251b773f1df21b29d7e4f98cd51fd3f445cc90_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/29d2db4ee28abc8ce03f9cc4b1251b773f1df21b29d7e4f98cd51fd3f445cc90_big.svg" type="image/svg+xml"></object> &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/08c4074084a7930b764f232ae54d300f2fd3d8fbe49360805bb4b1ebc509f4fe_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/08c4074084a7930b764f232ae54d300f2fd3d8fbe49360805bb4b1ebc509f4fe_big.svg" type="image/svg+xml"></object> &nbsp; <hr> <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/4767e380c61237501faffcb51449fdf08133bccd28e8833b9c3e72d3ad3f9724_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/4767e380c61237501faffcb51449fdf08133bccd28e8833b9c3e72d3ad3f9724_big.svg" type="image/svg+xml"></object> &nbsp;

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