Question

11 Fie A, B = C(0,r) astfel încât m(*AOB) = uº. Determinați lungimea arcului AB şi aria sectorului cerc AOB dacă:
a r 6 cm și u° = 45°;
c r=2 cm şi u° = 120°;
b r=4 cm şi uº=30°;
d 'r = 10 cm şi u° = 90°.​

Answer 1

Ai toate informațiile necesare, trebuie doar să aplici formulele,

$\displaystyle L{_{arc}} =\frac{\pi\cdot r\cdot u^{\circ}}{180^{\circ}}$

$\displaystyle A{_{sector} } =\frac{\pi\cdot r^{2} \cdot u^{\circ}}{360^{\circ}}$

a) r = 6 cm și u° = 45°

$\displaystyle L{_{arc}} =\frac{\pi\cdot r\cdot u^{\circ}}{180^{\circ}}=\frac{\pi\cdot 6\cdot 45^{\circ}}{180^{\circ}}=\frac{\pi\cdot 45^{\circ}}{30^{\circ}}=\frac{3\pi}{2}\ cm$

$\displaystyle A{_{sector} } =\frac{\pi\cdot r^{2} \cdot u^{\circ}}{360^{\circ}}=\frac{\pi\cdot 6^{2} \cdot 45^{\circ}}{360^{\circ}}=\frac{\pi \cdot 45^{\circ}}{10^{\circ}}=\frac{9\pi }{2}\ cm^{2}$

b) r = 4 cm şi uº = 30°

$\displaystyle L{_{arc}} =\frac{\pi\cdot r\cdot u^{\circ}}{180^{\circ}}=\frac{\pi\cdot 4\cdot 30^{\circ}}{180^{\circ}}=\frac{4\pi}{6}=\frac{2\pi}{3}\ cm$

$\displaystyle A{_{sector} } =\frac{\pi\cdot r^{2} \cdot u^{\circ}}{360^{\circ}}=\frac{\pi\cdot 4^{2} \cdot 30^{\circ}}{360^{\circ}}=\frac{16\pi}{12}=\frac{4\pi }{3}\ cm^{2}$

c) r = 2 cm şi u° = 120°

$\displaystyle L{_{arc}} =\frac{\pi\cdot r\cdot u^{\circ}}{180^{\circ}}=\frac{\pi\cdot 2\cdot 120^{\circ}}{180^{\circ}}=\frac{2\pi\cdot 2}{3}=\frac{4\pi}{3}\ cm$

$\displaystyle A{_{sector} } =\frac{\pi\cdot r^{2} \cdot u^{\circ}}{360^{\circ}}=\frac{\pi\cdot 2^{2} \cdot 120^{\circ}}{360^{\circ}}=\frac{4\pi}{3}\ cm^{2}$

d) r = 10 cm şi u° = 90°

$\displaystyle L{_{arc}} =\frac{\pi\cdot r\cdot u^{\circ}}{180^{\circ}}=\frac{\pi\cdot 10\cdot 90^{\circ}}{180^{\circ}}=\frac{10\pi}{2}=5\pi\ cm$

$\displaystyle A{_{sector} } =\frac{\pi\cdot r^{2} \cdot u^{\circ}}{360^{\circ}}=\frac{\pi\cdot 10^{2} \cdot 90^{\circ}}{360^{\circ}}=\frac{100\pi}{4}=25\pi\ cm^{2}$