A circle has a chord that goes from $\frac{π}{8}$ $( pi)/8$ to $\frac{4 π}{3}$ $(4 pi) / 3$ radians on the circle. If the area of the circle is $75 π$ $75 pi$ , what is the length of the chord?

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A circle has a chord that goes from $\frac{π}{8}$ $( pi)/8$ to $\frac{4 π}{3}$ $(4 pi) / 3$ radians on the circle. If the area of the circle is $75 π$ $75 pi$ , what is the length of the chord?

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Answer 1 · 2017-09-07T13:00:59

$16.4 units$ $16.4" units"$

Explanation:

$calculate the length of the chord using the cosine rule$ $"calculate the length of the chord using the "color(blue)"cosine rule"$

$color(red)(bar(ul(|color(white)(2/2)color(black)(c^2=a^2+b^2-(2abcosC)color(white)(2/2)|)))$

$"where c is the length of the chord, a and b are the radii"$
$"and C is the angle subtended at the centre of the circle"$
$"by the chord"$

$pir^2=75pilarr" area of circle"$

$rArrr^2=75rArrr=+-sqrt75=5sqrt3$

$angleC=(4pi)/3-pi/8=(32pi)/24-(3pi)/24=(29pi)/24$

$c^2=(5sqrt3)^2+(5sqrt3)^2-(2xx5sqrt3xx5sqrt3xxcos((29pi)/24))$

$color(white)(c^2)=75+75-(-119)=269$

$rArrc=sqrt269~~ 16.4" units"$

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A circle has a chord that goes from $\frac{π}{8}$ $( pi)/8$ to $\frac{4 π}{3}$ $(4 pi) / 3$ radians on the circle. If the area of the circle is $75 π$ $75 pi$ , what is the length of the chord?

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A circle has a chord that goes from ( pi)/8 π8( pi)/8 to (4 pi) / 3 4π3(4 pi) / 3 radians on the circle. If the area of the circle is 75 pi 75π75 pi , what is the length of the chord?

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A circle has a chord that goes from $\frac{π}{8}$ $( pi)/8$ to $\frac{4 π}{3}$ $(4 pi) / 3$ radians on the circle. If the area of the circle is $75 π$ $75 pi$ , what is the length of the chord?