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jasenka [17]
2 years ago
8

Find the area of the shaded region

Mathematics
1 answer:
Alex777 [14]2 years ago
7 0

∆BOC is equilateral, since both OC and OB are radii of the circle with length 4 cm. Then the angle subtended by the minor arc BC has measure 60°. (Note that OA is also a radius.) AB is a diameter of the circle, so the arc AB subtends an angle measuring 180°. This means the minor arc AC measures 120°.

Since ∆BOC is equilateral, its area is √3/4 (4 cm)² = 4√3 cm². The area of the sector containing ∆BOC is 60/360 = 1/6 the total area of the circle, or π/6 (4 cm)² = 8π/3 cm². Then the area of the shaded segment adjacent to ∆BOC is (8π/3 - 4√3) cm².

∆AOC is isosceles, with vertex angle measuring 120°, so the other two angles measure (180° - 120°)/2 = 30°. Using trigonometry, we find

\sin(30^\circ) = \dfrac{h}{4\,\rm cm} \implies h= 2\,\rm cm

where h is the length of the altitude originating from vertex O, and so

\left(\dfrac b2\right)^2 + h^2 = (4\,\mathrm{cm})^2 \implies b = 4\sqrt3 \,\rm cm

where b is the length of the base AC. Hence the area of ∆AOC is 1/2 (2 cm) (4√3 cm) = 4√3 cm². The area of the sector containing ∆AOC is 120/360 = 1/3 of the total area of the circle, or π/3 (4 cm)² = 16π/3 cm². Then the area of the other shaded segment is (16π/3 - 4√3) cm².

So, the total area of the shaded region is

(8π/3 - 4√3) + (16π/3 - 4√3) = (8π - 8√3) cm²

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olganol [36]

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given,

on first stop

number of car = 20  and number of trucks = 18

on second stop

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we need to calculate which rest stop has higher ratio of car to truck.

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ratio= r₁ =\dfrac{cars}{trucks}

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Rest Stop 2

ratio= r₂ =\dfrac{cars}{trucks}

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3 0
3 years ago
A student determined that the area of the segment of c shown above is Asegment = 137.71 ft2. The student's work is shown below.
ivolga24 [154]

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Option D. The student did not use the correct formula to calculate the area of the segment

Step-by-step explanation:

step 1

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Applying the law of sines

A=\frac{1}{2}(12^{2})sin(60\°)=62.35\ ft^{2}

step 2

Find the area of the sector

The area of the sector is 1/6 of the area of the circle

so

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substitute the value

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step 3

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The area of the segment is equal to the area of sector minus the area of triangle

A=75.36\ ft^{2}-62.35\ ft^{2}=13.01\ ft^{2}

therefore

The student did not use the correct formula to calculate the area of the segment

4 0
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