Explanation:
The formula to find the arc length of a circle is:
arc length = (central angle / 360 degrees) x 2πr
where r is the radius of the circle.
In this case, we are given the central angle of the arc, which is 36 degrees, and the length of the chord AB, which is 3 units. To find the radius of the circle, we need to use some trigonometry.
Since angle ABC is an inscribed angle that intercepts arc AC, it is half the measure of arc AC. So, we can find the measure of arc AC as 2 times the measure of angle ABC, which is:
m(arc AC) = 2 x m∠ABC
m(arc AC) = 2 x 36
m(arc AC) = 72 degrees
Now, let's use the law of cosines to find the radius of the circle:
c^2 = a^2 + b^2 - 2ab cos(C)
where a and b are the sides of the triangle and c is the side opposite the angle C.
In triangle ABC, we know that side AB has length 3 units and angle ABC has measure 36 degrees. We also know that angle BAC is a right angle, since it is formed by a chord and a radius of a circle. So, we can use trigonometry to find the length of side AC:
sin(36) = AC / 2r
2r = AC / sin(36)
r = AC / (2 sin(36))
Using the law of cosines, we can find AC:
c^2 = a^2 + b^2 - 2ab cos(C)
AC^2 = 3^2 + (2r)^2 - 2(3)(2r) cos(36)
AC^2 = 9 + 4r^2 - 12r cos(36)
Substituting the expression for r from above, we get:
AC^2 = 9 + 4/((2 sin(36))^2) - 12/((2 sin(36))) cos(36)
AC^2 = 9.259
Taking the square root of both sides, we get:
AC ≈ 3.04 units
Now that we know the radius of the circle and the central angle of the arc, we can use the formula for arc length to find the length of arc AC:
arc length = (central angle / 360 degrees) x 2πr
arc length = (72 / 360) x 2π(3.04)
arc length ≈ 3.03 units
Therefore, the length of arc AC is approximately 3.03 units, rounded to the nearest hundredth.