A ship is passing through the island of corregidor. at its closest point of approach, Corregidor radar determines that it is 2, 400 meters away. Later,
the radar determines that it is 2, 650 meters away.
a.) By what angle did the ship's bearing from corregidor change?
b.) How far did the Ship travel between the two observation points?
Trigonometry
To calculate the angle and distance, we can use basic trigonometry.
a.) To find the change in bearing angle, we can use the formula:
change in bearing angle = arccos((d2 - d1) / r)
where d1 and d2 are the distances from Corregidor at the closest and farthest points, and r is the radius of the Earth.
In this case, d1 = 2400 meters and d2 = 2650 meters. We need to convert these distances to radians, so:
d1_rad = d1 / r
d2_rad = d2 / r
The radius of the Earth, r, is approximately 6371 kilometers or 6,371,000 meters. Therefore:
d1_rad = 2400 / 6,371,000
≈ 0.000376819 radians
d2_rad = 2650 / 6,371,000
≈ 0.000416022 radians
Now, we can calculate the change in bearing angle:
change in bearing angle = arccos(d2_rad - d1_rad)
= arccos(0.000416022 - 0.000376819)
= arccos(0.000039203)
≈ 0.00224 radians
To convert this angle to degrees, we can multiply by 180/π:
change in bearing angle (in degrees) ≈ 0.00224 * (180 / π)
≈ 0.128 degrees
Therefore, the ship's bearing from Corregidor changed approximately 0.128 degrees.
b.) To find the distance traveled by the ship between the two observation points, we can use the formula:
distance = 2 * r * sin(change in bearing angle / 2)
Given the change in bearing angle is 0.00224 radians, and using the same value for r as above:
distance = 2 * 6,371,000 * sin(0.00224 / 2)
= 12,742,000 * sin(0.00112)
≈ 16,855 meters
Therefore, the ship traveled approximately 16,855 meters between the two observation points.
To solve this problem, we can use the concept of trigonometry and apply it to the given information. Let's break down the problem into two parts:
a) To find the angle by which the ship's bearing from Corregidor changed, we will use the tangent function. The tangent of an angle is equal to the ratio of the opposite side to the adjacent side.
Given that the ship's distance from Corregidor changed from 2400 meters to 2650 meters, we can calculate the change in the opposite side. It can be found by subtracting the initial distance from the final distance:
Change in opposite side = 2650 meters - 2400 meters = 250 meters.
Now, taking the change in the opposite side as the numerator and the adjacent side as the denominator, we can find the tangent of the angle:
Tangent of the angle = (Change in opposite side) / (Adjacent side).
However, we need to find the adjacent side. Since the ship is always moving away, we can take the initial distance as the adjacent side. Hence:
Adjacent side = 2400 meters.
Now, substituting the values into the formula, we get:
Tangent of the angle = 250 meters / 2400 meters.
To find the angle itself, we will take the inverse tangent (arctan) of the above value:
Angle = arctan(250 meters / 2400 meters).
Using a scientific calculator or an online calculator, you can find the value of the angle by entering the above expression.
b) To determine how far the ship traveled between the two observation points, we need to calculate the distance between the initial and final points. This can be done by subtracting the initial distance from the final distance:
Distance traveled = Final distance - Initial distance.
Given that the initial distance is 2400 meters and the final distance is 2650 meters, we have:
Distance traveled = 2650 meters - 2400 meters.
Simply subtracting these values will give you the distance traveled by the ship between the two observation points.
at CPA, the distance is 90 degrees frm the course.
So you have right angle triangle.
angle=arccos(2400/2650)
distancetraveled= 2650*sinAngle