Question

A circle's center is at `(4 ,7 )` and it passes through `(9 ,1 )`. What is the length of an arc covering `(5pi ) /3` radians on the circle?

Answer 1

Length of the Arc`~=41`

Explanation:

Circle's center is at `(4,7)` and it passes through `(9,1)`
Therefore Length of the `radius=r` =Distance between these points`(4,7) and (9,1)`
or
`radius =r=sqrt((9-4)^2+(7-1)^2)`
`=sqrt(5^2+6^2)`
`=sqrt(25+36)`
`=sqrt61`
`=7.81`
Therefore Circumfernce of the Circle `=2pir=2pitimes7.81~=49`
Arc covers `(5pi)/3` radians on the Circle
In other words Arc covers `(5pi)/3-:2pi=5/6times`(circumference of the Circle)
Therefore Length of the Arc `=5/6 times2pir=5/6 times49~=41`