Final answer:
The measure of the arc on the tree that the hummingbird can see is approximately 5.67 feet.
Step-by-step explanation:
To find the measure of the arc on the tree that the hummingbird can see, we need to use the concept of tangents. When the hummingbird is 33 feet from the center of the tree, the two lines of sight it forms with the tree create two tangents. Let's consider the right triangle formed by the hummingbird, the center of the tree, and the point of tangency on the tree.
We can use the Pythagorean theorem to find the length of the line segment connecting the hummingbird to the center of the tree, which is the radius. The radius is 6 feet. Next, we can use trigonometry to determine the measure of the angle formed by the line segment connecting the hummingbird to the point of tangency and the radius. The tangent of this angle will be the ratio of the side opposite the angle (the arc length) to the side adjacent to the angle (the distance between the hummingbird and the center of the tree).
The measure of the arc on the tree that the hummingbird can see can be found by multiplying the radius of the tree by the tangent of the angle. Let's calculate it:
tan(θ) = (opposite/adjacent) = (arc length)/(distance between the hummingbird and the center of the tree)
arc length = tan(θ) * radius of the tree
arc length = tan(θ) * 6 feet
Now, we need to find the measure of the angle θ. We can use the inverse tangent function to calculate it:
θ = atan(distance between the hummingbird and the center of the tree / radius of the tree)
θ = atan(33 feet / 6 feet)
Now, we can substitute the value of θ into the equation for arc length:
arc length = tan(atan(33 feet / 6 feet)) * 6 feet
arc length ≈ 0.945 * 6 feet
arc length ≈ 5.67 feet