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A chord with a length of #13 # runs from #pi/12 # to #pi/2 # radians on a circle. What is the area of the circle?

Answer 1

Evelyn Arboleda

#color(brown)("Area of the circle " A_c = pi r^2 = pi * (10.68)^2 = 358.34 " sq units"#

#"Given ; " hat (AOB) = pi/2 - pi/12 = (5pi)/12, vec(AB) = 13#

#hat (AOM) = hat (AOB) / 2 = (5pi)/24#

#bar(OA) = r = bar(AM) / sin (AOM) = (bar(AB)/2) / sin (AOM)#

#r = (13/2) / sin ((5pi)/24) = 10.68 " units"#

#"Area of the circle " A_c = pi r^2 = pi * (10.68)^2 = 358.34 " sq units"#

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Answer 2

James Atkins

To find the area of the circle, we need to first determine its radius. Given that the chord length is 13 and it subtends an arc from π/12 to π/2 radians, we can use the formula for the length of a chord in a circle:

chord length = 2 * radius * sin(θ/2)

where θ is the central angle subtended by the chord.

Substituting the given values:

13 = 2 * radius * sin((π/2 - π/12)/2)

Solving for radius:

radius = 13 / (2 * sin(π/24))

Once we find the radius, we can use the formula for the area of a circle:

Area = π * radius^2

By plugging in the calculated value of the radius, we can find the area of the circle.

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