Answer to Question #125736 in Mechanics | Relativity for Ezeah Emmanuel Uchechukwu

Question #125736
1.An arc of a circle of radius 5cm subtend an angle of 102°at the center of circle calculate
a. the length of the arc
b. perimeter of the sector

2. The area of the sector which subtends an angle of x at the center of the circle of radius 5.2cm is 21cm square calculate:
a.the value of x
b.length of the arc
c. perimeter of the sector

3. The perimeter of the sector which subtends an angle of 105°at the center of a circle is 17.25m calculate:
a.radius of the circle
b.length of an arc and
c.area of the sector
1
Expert's answer
2020-07-10T10:27:44-0400

1a. The length of the circle arc is:


"L = \\pi r\\dfrac{\\theta}{180\\degree} = \\pi \\cdot 5\\dfrac{102\\degree}{180\\degree} \\approx8.9cm"

1b. Perimeter of the sector is the length of the circle arc and two bounding radii. Thus:


"P = L + 2r = 8.9+10 = 18.9 cm"

Answer. a) 8.9 cm, b) 18.9 cm.


2a. The area of sector is given by:


"A = \\pi r^2 \\dfrac{x}{360\\degree}"

Then


"x = \\dfrac{360\\degree A}{\\pi r^2} = \\dfrac{360\\degree\\cdot 21}{\\pi \\cdot 5.2^2}\\approx89\\degree"

2b. The length of the circle arc is:


"L = \\pi r\\dfrac{\\theta}{180\\degree} = \\pi \\cdot 5.2\\dfrac{89\\degree}{180\\degree} \\approx8.1cm"

2c. Perimeter of the sector is the length of the circle arc and two bounding radii. Thus:


"P = L + 2r = 8.1+10.4 = 18.5 cm"

Answer. a) 89 degrees, b) 8.1 cm, c) 18.5 cm.


3a. Perimeter of the sector is the length of the circle arc and two bounding radii:


"P = L + 2r"

Substituting the expression for "L", get:


"P = L + 2r = \\pi r\\dfrac{\\theta}{180\\degree} + 2r = r\\left(\\pi\\dfrac{\\theta}{180\\degree} + 2 \\right)"

Expressing "r", obtain:


"r = \\dfrac{P}{\\pi\\dfrac{\\theta}{180\\degree} + 2} = \\dfrac{17.25}{\\pi\\dfrac{105\\degree}{180\\degree} + 2} \\approx 4.5m"

3b. The length of the circle arc is:


"L = \\pi r\\dfrac{\\theta}{180\\degree} = \\pi\\cdot 4.5\\dfrac{105\\degree}{180\\degree} \\approx 8.2m"

3c. The area of the sector is:


"A = \\pi r^2 \\dfrac{\\theta}{360\\degree} = \\pi\\cdot 4.5^2 \\dfrac{105\\degree}{360\\degree} \\approx 18.6m^2"

Answer. a) 4.5m, b) 8.2m, c)18.6m^2.


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