a lighthouse A bears N 36 degree 20 minutes E at a ship S. after sailing 2.75 km in the direction N 78 degree 50 minutes E to point T, the bearing of the lighthouse becomes N 8 degree 30 minutes E. find the distance from A to T.
draw a diagram, and you can figure angle T. Then, use the law of sines.
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#junkterrorbill
To find the distance from point A to point T, we can use the concept of trigonometry and the information given in the problem.
1. Convert the given bearings from degrees and minutes to decimal degrees:
- N 36 degree 20 minutes E = 36.3333 degrees
- N 78 degree 50 minutes E = 78.8333 degrees
- N 8 degree 30 minutes E = 8.5 degrees
2. Draw a diagram to visualize the scenario:
- Place point A (the lighthouse) and point S (the ship) on the diagram with the given bearing of 36.3333 degrees.
- From point S, draw a line in the direction of N 78.8333 degrees E for a distance of 2.75 km. Mark the endpoint as point T.
- From point A, draw a line in the direction of N 8.5 degrees E (the new bearing after moving to point T).
3. Now, we have formed a triangle with sides of lengths 2.75 km, x km (the distance from A to T), and an unknown side that connects point A to point T. We need to find x, the distance from A to T.
4. Apply the law of sines to solve for x:
- sin(bearing from A to T) / side opposite the bearing = sin(90 degrees - angle at T) / side opposite the angle at T
- sin(8.5 degrees) / x = sin(90 degrees - 78.8333 degrees) / 2.75 km
5. Solve for x:
- x = (2.75 km * sin(8.5 degrees)) / sin(90 degrees - 78.8333 degrees)
Calculating this expression in degrees might lead to rounding errors, so it's better to convert the trigonometric functions to radians first:
- x = (2.75 km * sin(8.5*pi/180)) / sin((90 - 78.8333)*pi/180)
Using a calculator to evaluate this expression, we find that x is approximately 2.981 kilometers.
Therefore, the distance from point A to point T is approximately 2.981 kilometers.