Math  /  Geometry

QuestionThe diagram shows a sector of a circle radius 11 cm .
Show that the perimeter of the sect is greater than 47.5 cm . \square

Studdy Solution

STEP 1

1. The sector is part of a circle with a radius of 11 11 cm.
2. The central angle of the sector is 135 135^\circ .
3. The perimeter of the sector includes two radii and the arc length.

STEP 2

1. Calculate the arc length of the sector.
2. Calculate the perimeter of the sector.
3. Compare the perimeter to 47.5 47.5 cm.

STEP 3

Calculate the arc length of the sector using the formula for arc length:
Arc Length=θ360×2πr \text{Arc Length} = \frac{\theta}{360^\circ} \times 2\pi r
where θ=135 \theta = 135^\circ and r=11 r = 11 cm.
Arc Length=135360×2π×11 \text{Arc Length} = \frac{135}{360} \times 2\pi \times 11

STEP 4

Calculate the perimeter of the sector. The perimeter P P is the sum of the arc length and the two radii:
P=Arc Length+2r P = \text{Arc Length} + 2r
Substitute the arc length from Step 1 and r=11 r = 11 cm:
P=135360×2π×11+2×11 P = \frac{135}{360} \times 2\pi \times 11 + 2 \times 11

STEP 5

Calculate the numerical value of the perimeter:
First, calculate the arc length:
Arc Length=135360×2π×1125.929 cm \text{Arc Length} = \frac{135}{360} \times 2\pi \times 11 \approx 25.929 \text{ cm}
Then, calculate the perimeter:
P=25.929+22=47.929 cm P = 25.929 + 22 = 47.929 \text{ cm}

STEP 6

Compare the calculated perimeter to 47.5 47.5 cm:
Since 47.929 cm>47.5 cm 47.929 \text{ cm} > 47.5 \text{ cm} , the perimeter of the sector is greater than 47.5 47.5 cm.
The perimeter of the sector is indeed greater than 47.5 47.5 cm.

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