A circle's center is at #(2 ,4 )# and it passes through #(7 ,6 )#. What is the length of an arc covering #(15pi ) /8 # radians on the circle?

1 Answer
Alan P.
Nov 4, 2016

Length of the arc: #31.7# (approx.)

Explanation:

If the circle has a center at #(2,4)# and passes through #(7,6)# then it has a radius of
#color(white)("XXX")r=sqrt((7-2)^2+(6-4)^2)=sqrt(25+4) =sqrt(29)#
and a diameter of
#color(white)("XXX")d=2r=2sqrt(29)#

The circumference of the circle would be
#color(white)("XXX")"Circumference"_circ=pid = 2sqrt(29)pi#

The complete circle's circumference is covered by an arc of #2pi=(16pi)/8#

An arc of #(15pi)/8# will cover #((15pi)/8)/((16pi)/8)=15/16# of #"Circumference"_circ#

The given arc will cover
#color(white)("XXX")15/16xx2sqrt(29)pi#

#color(white)("XXX")~~31.72123912# (with a calculator)

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