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Sagot :
Answer:
300 cm²
Step-by-step explanation:
Since we don't know the measure of the central angle of sector CAT, we'll solve for it first.
We know that the formula in finding the arc length is
[tex]\sf Arc \: length = \frac{\theta}{360^{\circ}} 2\pi r[/tex]
Given,
- [tex]\sf Arc \: length = 60[/tex]
- [tex]\sf r = 10[/tex]
Substituting,
[tex]\sf 60 = \frac{\theta}{360^{\circ}} 2\pi \times 10[/tex]
[tex]\implies \sf 60 = \frac{\theta}{360^{\circ}} 20\pi[/tex]
[tex]\implies \sf \frac{60(360^{\circ})}{20\pi} = \theta[/tex]
[tex]\implies \sf \frac{3(360^{\circ})}{\pi} = \theta[/tex]
[tex]\implies \sf \frac{1080^{\circ}}{\pi}= \theta[/tex]
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We already solved the central angle of arc CT/sector CAT, which means that we can now solve for the area of sector CAT.
The formula in calculating the area of a sector is
[tex]\sf Area \: of \: sector = \frac{\theta}{360^{\circ}} \pi r^2[/tex]
Substituting,
[tex]\sf Area \: of \: sector =\frac{\frac{1080^{\circ}}{\pi} }{360^{\circ}}\pi \times 10^2[/tex]
[tex]\implies \sf Area \: of \: sector = \frac{1080^{\circ}}{\pi} \times \frac{1}{360^{\circ}} \times 100\pi[/tex]
[tex]\implies \sf Area \: of \: sector = \frac{1080^{\circ}\times 100\pi}{360^{\circ}\pi}[/tex]
[tex]\implies \sf Area \: of \: sector = \frac{300\pi}{\pi}[/tex]
[tex]\implies \boxed{\boxed{\sf Area \: of \: sector= 300}}[/tex]
Therefore, the area of sector CAT is 300 cm²
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