A park maiand mantenance person stands 16m from a circular monument. Assume that her lines of sight from tangents to the monument and make an angle of 47 degrees. what is the measure of the arc of the monument that her lines of sight intersected
possible answers
a: 23.5 degrees
b: 94 degrees
c: 266 degrees
d: 133 degrees
See 5-3-12,5:01PM post.
133
To solve this problem, we need to use the concept that the angle between a tangent line and a radius of a circle is 90 degrees.
Given:
- The park maintenance person stands 16 meters from a circular monument.
- Her lines of sight from tangents to the monument make an angle of 47 degrees.
We need to find the measure of the arc of the monument that her lines of sight intersected.
Since the angle between a radius and a tangent is 90 degrees, we can subtract the given angle of 47 degrees from 90 degrees to find the angle of the arc intersected by the lines of sight.
90 degrees - 47 degrees = 43 degrees
Therefore, the measure of the arc of the monument that her lines of sight intersected is 43 degrees.
None of the given answer choices match this result, so it seems there might be a mistake either in the question or the answer choices.
To find the measure of the arc of the monument that the lines of sight intersected, we can use the fact that the angle between the tangent and the radius of a circle is equal to the angle formed by these two tangents. In this case, we know that the angle between the tangents is 47 degrees.
First, let's draw a diagram to visualize the situation.
We have a circular monument, and the park maintenance person stands 16m away from it. The lines of sight from the person form an angle of 47 degrees.
Now, consider the triangle formed by the center of the circle, the point where the person is standing, and one of the points where the lines of sight intersect the circle. This is a right triangle because the line from the center to the point on the circle is perpendicular to the tangent line.
Since the person is standing 16m away from the monument, we have one side of the triangle. Let's call this side "adjacent." The other side of the triangle is the radius of the circle, which we'll call "opposite." We need to find the angle opposite to the side with length 16m, which is the angle formed by the two lines of sight.
Using trigonometry, we can use the tangent function. The tangent of an angle is equal to the ratio of the length of the opposite side to the adjacent side.
tan(angle) = opposite/adjacent
In this case, we have:
tan(angle) = opposite/16
We can solve for the opposite side by rearranging the equation:
opposite = tan(angle) * 16
Now, we can calculate the length of the opposite side:
opposite = tan(47) * 16
Using a calculator, we find that tan(47) is approximately 1.0724. Thus, we have:
opposite = 1.0724 * 16
opposite = 17.1576m (approximately)
Now, we know the length of the opposite side, which is the radius of the circle. The arc of a circle with a given central angle is, by definition, the length of the circumference of the circle multiplied by the ratio of the central angle to 360 degrees.
The circumference of a circle is given by the formula:
circumference = 2π * radius
So, in this case, we have:
arc = (47/360) * (2 * π * 17.1576)
arc ≈ (47/360) * (34.3152 * π)
arc ≈ 4.4055 * π
To find the measure of the arc in degrees, we need to convert it from radians to degrees:
arc ≈ 4.4055 * (180/π) ≈ 253.09 degrees
Since none of the given answer choices match the calculated measure of the arc (which is approximately 253.09 degrees), it seems there may be an error in the problem or the answer choices given.