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МатематикаРешебник по алгебре и начала математического анализа 10 класс Колягин Задание 933
Задание 933
\[\boxed{\mathbf{933.}}\]
| \[Угол,\ {^\circ}\] | \[30\] | \[36\] | \[\frac{90}{\pi}\] | \[\frac{720}{\pi}\] | \[\frac{360}{\pi}\] | \[\frac{180}{\pi}\] |
|---|---|---|---|---|---|---|
| \[Угол,\ рад\] | \[\frac{\pi}{6}\] | \[\frac{\pi}{5}\] | \[\frac{1}{2}\] | \[4\] | \[2\] | \[1\] |
| \[Радиус,\ см\] | \[2\] | \[\frac{10}{\pi}\] | \[10\] | \[5\] | \[5\] | \[10\] |
| \[Длина\ дуги,\ см\] | \[\frac{\pi}{3}\] | \[2\] | \[5\] | \[20\] | \[10\] | \[10\] |
| \[Площадь\ сектора,\ см^{2}\] | \[\frac{\pi}{3}\] | \[\frac{10}{\pi}\] | \[25\] | \[50\] | \[25\] | \[50\] |
\[\mathbf{Первый\ столбец:}\]
\[a = 30{^\circ} = \frac{\pi \bullet 30}{180} = \frac{\pi}{6}\]
\[l = aR = \frac{\pi}{6} \bullet 2 = \frac{\pi}{3}\]
\[S = \frac{R^{2}}{2} \bullet a = \frac{2^{2}}{2} \bullet \frac{\pi}{6} = 2 \bullet \frac{\pi}{6} = \frac{\pi}{3}\]
\[Второй\ столбец:\]
\[a = \frac{\pi}{5} = \left( \frac{180}{\pi} \bullet \frac{\pi}{5} \right)^{{^\circ}} = 36{^\circ}\]
\[R = \frac{l}{a} = 2\ :\frac{\pi}{5} = 2 \bullet \frac{5}{\pi} = \frac{10}{\pi}\]
\[S = \frac{R^{2}}{2} \bullet a = \left( \frac{10}{\pi} \right)^{2}\ :2 \bullet \frac{\pi}{5} =\]
\[= \frac{100}{2\pi^{2}} \bullet \frac{\pi}{5} = \frac{10}{\pi}\]
\[Третий\ столбец:\]
\[a = \frac{l}{R} = \frac{5}{10} = \frac{1}{2}\]
\[a = \left( \frac{180}{\pi} \bullet \frac{1}{2} \right)^{{^\circ}} = \left( \frac{90}{\pi} \right)^{{^\circ}}\]
\[S = \frac{R^{2}}{2} \bullet a = \frac{10^{2}}{2} \bullet \frac{1}{2} = \frac{100}{4} = 25\]
\[Четвертый\ столбец:\]
\[a = \frac{2S}{R^{2}} = \frac{2 \bullet 50}{5^{2}} = \frac{100}{25} = 4\]
\[a = \left( \frac{180}{\pi} \bullet 4 \right)^{{^\circ}} = \left( \frac{720}{\pi} \right)^{{^\circ}}\]
\[l = aR = 4 \bullet 5 = 20\]
\[Пятый\ столбец:\]
\[a = \left( \frac{180}{\pi} \bullet 2 \right)^{{^\circ}} = \left( \frac{360}{\pi} \right)^{{^\circ}}\]
\[R = \sqrt{\frac{2S}{a}} = \sqrt{\frac{2 \bullet 25}{2}} = \sqrt{25} = 5\]
\[l = aR = 2 \bullet 5 = 10\]
\[Шестой\ столбец:\]
\[R = \frac{l}{a}\]
\[S = \frac{R^{2}}{2} \bullet a = \frac{l^{2}}{2a^{2}} \bullet a = \frac{l^{2}}{2a}\text{\ \ }\]
\[a = \frac{2S}{l^{2}} = \frac{2 \bullet 50}{10^{2}} = 1\]
\[a = \left( \frac{180}{\pi} \right)^{{^\circ}}\]
\[R = 10.\]
