Tambayoyi Misali Game da Alaƙar Tsakanin Tsawon Arc da Yankin Sashe
A cikin darussa na lissafi, musamman a cikin nazarin da'irori, sau da yawa muna cin karo da ra'ayoyin tsawon baka da yankin sashe. Waɗannan ra'ayoyi guda biyu suna da mahimmanci don fahimtar abubuwan da suka shafi lissafi daban-daban da suka shafi da'irori. Bari mu fara bayyana waɗannan ra'ayoyi guda biyu kafin mu ba da misalai na matsalolin da mafitarsu.
Tsawon Baka
Tsawon baka shine nisan da ke kan baka tsakanin maki biyu a kan da'ira. Don ƙididdige tsawon baka na da'ira, yawanci muna buƙatar radius na da'irar (r) da kusurwar tsakiya (θ) wanda baka ke ƙarƙashinsa a cikin radians. Ana iya rubuta dabarar ƙididdige tsawon baka (s) kamar haka:
\[s = r \times \theta \]
Idan an ba da kusurwar tsakiya a digiri, dole ne mu fara canza ta zuwa radians ta hanyar:
\[ \theta_{radian} = \theta_{degree} \times \frac{\pi}{180} \]
Yankin Sashe
Sashe wani ɓangare ne na da'ira da aka ɗaure da radius biyu da baka tsakanin su. Don ƙididdige yankin wani sashe, muna amfani da radius na da'irar (r) da kusurwar tsakiya (θ). Tsarin ƙididdige yankin wani sashe (A) shine:
\[ A = \frac{1}{2} r^2 \times \theta \]
Kamar yadda yake a tsawon baka, idan an auna kusurwar tsakiya da digiri, dole ne mu fara mayar da ita zuwa radians.
Tambayoyi da Tattaunawa Samfura
Domin fayyace manufar tsawon baka da yankin sashe, bari mu sake duba tambayoyin misalai masu zuwa da tattaunawarsu.
Tambaya ta 1:
Idan aka ba da da'ira mai radius na 10 cm da kusurwar tsakiya ta digiri 60, ƙididdige tsawon baka da yankin ɓangaren da kusurwar ta samar.
Tattaunawa:
1. Lissafin Tsawon Baka:
– Da farko, muna canza kusurwar daga digiri zuwa radians:
\[ \theta = 60 \times \frac{\pi}{180} = \frac{\pi}{3} \, \text{radian} \]
– Ta amfani da dabarar tsawon baka:
\[s = r \times \theta \]
\[s = 10 \sau \frac{\pi}{3} \]
\[ s = \frac{10\pi}{3} \, \text{cm} \]
2. Lissafin Yankin Sashe:
– Amfani da dabarar fannin wani fanni:
\[ A = \frac{1}{2} r^2 \times \theta \]
\[ A = \frac{1}{2} \sau 10^2 \sau \frac{\pi}{3} \]
\[ A = \frac{1}{2} \sau 100 \sau \frac{\pi}{3} \]
\[ A = \frac{100\pi}{6} \]
\[ A = \frac{50\pi}{3} \, \text{cm}^2 \]
Don haka, tsawon baka shine \(\frac{10\pi}{3}\) cm, kuma yankin ɓangaren shine \(\frac{50\pi}{3}\) cm².
Tambaya ta 2:
Da'ira tana da radius na 7 cm da kusurwar tsakiya da ke ƙarƙashin baka na radian guda 2. Ka ƙayyade tsawon baka da yankin ɓangaren da'irar.
Tattaunawa:
1. Lissafin Tsawon Baka:
– Kusurwar tsakiya ta riga ta kasance a cikin radians, don haka za mu iya amfani da dabarar tsawon baka kai tsaye:
\[s = r \times \theta \]
\[s = 7 \sau 2 \]
\[s = 14 \, \rubutu{cm} \]
2. Lissafin Yankin Sashe:
– Amfani da dabarar fannin wani fanni:
\[ A = \frac{1}{2} r^2 \times \theta \]
\[ A = \frac{1}{2} \sau 7^2 \sau 2 \]
\[ A = \frac{1}{2} \sau 49 \sau 2 \]
\[ A = 49 \, \rubutu{cm}^2 \]
Don haka, tsawon baka shine 14 cm, kuma yankin ɓangaren shine 49 cm².
Tambaya ta 3:
Da'ira mai radius na 12 cm tana da sashe wanda tsawon bakansa shine 15 cm. Kayyade kusurwar tsakiya a digiri da yankin ɓangaren.
Tattaunawa:
1. Ƙayyade Kusurwar Tsakiya:
- Ta amfani da dabarar tsawon baka don nemo kusurwar tsakiya:
\[s = r \times \theta \]
\[ 15\pi = 12 \sau \theta \]
\[ \theta = \frac{15\pi}{12} \]
\[ \theta = \frac{5\pi}{4} \, \text{radian} \]
- Canza kusurwar tsakiya zuwa digiri:
\[ \theta = \frac{5\pi}{4} \times \frac{180}{\pi} \]
\[ \theta = \frac{5 \times 180}{4} \]
\[ \theta = 225 \, \text{digiri} \]
2. Lissafin Yankin Sashe:
– Amfani da dabarar fannin wani fanni:
\[ A = \frac{1}{2} r^2 \times \theta \]
\[ A = \frac{1}{2} \sau 12^2 \sau \frac{5\pi}{4} \]
\[ A = \frac{1}{2} \sau 144 \sau \frac{5\pi}{4} \]
\[ A = sau 72 \frac{5\pi}{4} \]
\[ A = 90\pi \, \rubutu{cm}^2 \]
Don haka, kusurwar tsakiya ta ɓangaren ita ce digiri 225, kuma yankin ɓangaren shine 90\(pi\) cm².
Kammalawa
Fahimtar alaƙar da ke tsakanin tsawon baka da yankin sashe yana buƙatar cikakken fahimtar ƙa'idodin asali na da'ira da kuma amfani da dabarun da suka dace. Ta hanyar matsalolin aiki da ke sama, za mu iya ganin mahimmancin sarrafa jujjuyawar kusurwa da kuma amfani da dabarun kai tsaye a cikin mahallin yanayin lissafi na zagaye. Kowane mataki a cikin tattaunawar matsaloli yana taimaka mana mu fahimci yadda dabarun ke aiki da kuma yadda za mu yi amfani da su yadda ya kamata.
Ta hanyar ci gaba da yin aiki da fahimtar muhimman abubuwan da aka bayyana, za mu ƙara ƙwarewa wajen magance matsalolin da suka shafi tsawon baka da yankin sashe, kuma wannan zai yi amfani sosai a fannoni daban-daban na lissafi da sauran aikace-aikacen kimiyya.
