Question

A circle's center is at `(3 ,1 )` and it passes through `(6 ,6 )`. What is the length of an arc covering `(13pi ) /12` radians on the circle?

Answer 1

`19.845`

Explanation:

The radius of the circle is
`color(white)("XXX")r=sqrt((6-3)^2+(6-1)^2) = sqrt(3^2+5^2) = sqrt(34)`

Since a circle has a circumference (arc length) of `2rpi`
and an arc angle of `2pi`:
`("arc length")/("arc angle") = (2rpi)/(2pi) = ("required arc length")/((13pi)/12)`

`"required arc length"=(13pi)/12*(cancel(2)rcancel(pi))/(cancel(2)(cancelpi))=(13pir)/12`

`color(white)("XXXXXXXXXXX")=(13sqrt(34)pi)/12`

`color(white)("XXXXXXXXXXX")~~19.845` (using a calculator)