Question

LINGKARAN yang berjari-jari 14 cm dan berpusat di O mempunyai panjang TALI BUSUR AB= 7√2 cm. Tentukan Luas Juring AOB ?
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Answer 1

Jawaban:

luas juring = 77 cm²

Penjelasan dengan langkah-langkah:

tali busur = 2r sin(a/2)

[tex]7 \sqrt{2} = 2(14) \: \sin \frac{\alpha}{2}[/tex]

[tex]7 \sqrt{2} = 28 \: \sin \frac{\alpha}{2}[/tex]

[tex] \displaystyle \sin \alpha = \frac{7 \sqrt{2} }{28} \times 2[/tex]

[tex] \displaystyle \sin \alpha = \frac{1 \sqrt{2} }{4} \times 2[/tex]

[tex] \displaystyle \sin \alpha = \frac{2 \sqrt{2} }{4} [/tex]

[tex]\displaystyle \sin \alpha = \frac{\sqrt{2} }{2} [/tex]

[tex]\displaystyle \sin \alpha \degree = \displaystyle \sin 45 \degree [/tex]

luas juring:

[tex] = \displaystyle \frac{ \alpha}{360 \degree} \times( \pi \times {r}^{2} )[/tex]

[tex] = \displaystyle \frac{ 45 \degree}{360 \degree} \times( \frac{22}{7} \times {14}^{2} )[/tex]

[tex] = \displaystyle \frac{ 1 \degree}{8 \degree} \times( 22 \times 14 \times 2)[/tex]

[tex] = \displaystyle \frac{ 1 \degree}{8 \degree} \times(616 \: {cm}^{2} )[/tex]

[tex] = \displaystyle 77 \: {cm}^{2} [/tex]

luas juring = 77 cm²

Answer 2

Verified answer

Jawab:

70,84 cm²

Penjelasan dengan langkah-langkah:

Tentukan cos ∠AOB dengan aturan cosinus

[tex]\begin{aligned}AB^2&\:=OA^2+OB^2-2~OB~ AB\cos\angle AOB\\\left ( 7\sqrt{2} \right )^2\:&=14^2+14^2-2(14)(14)\cos\angle AOB\\98\:&=392-392\cos\angle AOB\\\cos\angle AOB\:&=\frac{3}{4}\\\angle AOB\approx41,40^\circ\end{aligned}[/tex]

Perhitungan arccos menggunakan kalkulator ilmiah

Hitung luas juring AOB

[tex]\begin{aligned}L~AOB&\:=\frac{\angle AOB}{360^\circ}~\pi r^2\\\:&=\frac{41,40^\circ}{360^\circ}\left ( \frac{22}{7} \right )(14)^2\\\:&=70,84~\mathrm{cm^2}\end{aligned}[/tex]

Cara lain

Berdasarkan rumus panjang busur lingkaran AB = 2r sin (½ ∠AOB)

[tex]\begin{aligned}AB&\:=2r\sin\left ( \frac{\angle AOB}{2}\right )\\7\sqrt{2}\:&=2(14)\sin\left ( \frac{\angle AOB}{2} \right )\\\sin\left ( \frac{\angle AOB}{2} \right )\:&=\frac{\sqrt{2}}{4}\end{aligned}[/tex]

maka

[tex]\begin{aligned}\cos\angle AOB&\:=1-2\sin^2\left ( \frac{\angle AOB}{2} \right )\\\:&=1-2\left ( \frac{\sqrt{2}}{4} \right )^2\\\:&=\frac{3}{4}\\\angle AOB\:&\approx 41,40^\circ\end{aligned}[/tex]

hitung luas juring AOB

[tex]\begin{aligned}L~AOB&\:=\frac{\angle AOB}{360^\circ}~\pi r^2\\\:&=\frac{41,40^\circ}{360^\circ}\left ( \frac{22}{7} \right )(14)^2\\\:&=70,84~\mathrm{cm^2}\end{aligned}[/tex]