a ship is stationary at sea a tugboat is 36 km away at a bearing of 130 and a yacht is 27 km from the tugboat at a bearing of 260 use a scale 1:450,000 to measure the distance from the ship to the yacht to the nearest km
To find the distance from the ship to the yacht, we can use the property of triangles and trigonometry. Here are the steps to solve the problem:
Step 1: Draw a diagram:
Draw a diagram to visualize the given information. Place the ship at the center, the tugboat 36 km away at a bearing of 130 degrees, and the yacht 27 km away from the tugboat at a bearing of 260 degrees.
Step 2: Identify the triangle:
From the diagram, we can see that we have a triangle formed by the ship, the tugboat, and the yacht.
Step 3: Find the missing side lengths of the triangle:
To find the distance from the ship to the yacht, let's label the missing side lengths of the triangle:
- Let x represent the distance from the ship to the yacht (which we need to find).
- The side connecting the ship and the tugboat is 36 km.
- The side connecting the tugboat and the yacht is 27 km.
Step 4: Use the law of cosines:
The law of cosines states that in a triangle ABC with side lengths a, b, and c, and angle C:
c^2 = a^2 + b^2 - 2ab * cos(C)
Applying the law of cosines to our triangle, we have:
x^2 = 36^2 + 27^2 - 2(36)(27) * cos(260 - 130)
Step 5: Solve for x:
x^2 = 1296 + 729 - 1944 * cos(130)
x^2 = 2025 - 1944 * (-0.6428)
x^2 = 2025 + 1248.06432
x^2 = 3273.06432
Taking the square root of both sides, we get:
x = √3273.06432
x ≈ 57.2 km
Step 6: Convert the distance to the nearest kilometer:
Using the given scale of 1:450,000, we can convert the distance from kilometers to units on the map. Since 1 cm on the map represents 450,000 cm in reality, we can set up the proportion:
1 cm / 450,000 cm = x km / 57.2 km
Solving for x km:
x km = (1 cm / 450,000 cm) * 57.2 km
x km ≈ 0.000127 km
Finally, converting the result to the nearest kilometer, we get:
The distance from the ship to the yacht is approximately 0 kilometers.
To find the distance from the ship to the yacht, we can use the concept of vector addition. Here's how you can calculate it step by step:
Step 1: Draw a diagram representing the situation described.
Draw a point representing the ship, a point representing the tugboat, and a point representing the yacht, in the correct relative positions based on the given information. Label the distances between them as mentioned: 36 km between the ship and the tugboat and 27 km between the tugboat and the yacht.
Step 2: Determine the direction of each bearing.
The bearing of 130 degrees means the angle between the reference direction (usually north) and the direction of the tugboat from the ship is 130 degrees. Similarly, the bearing of 260 degrees means the angle between the reference direction and the direction of the yacht from the tugboat is 260 degrees.
Step 3: Convert the bearings to angles relative to the positive x-axis.
To calculate the angle relative to the positive x-axis, subtract the given bearing angle from 90 degrees. For the tugboat, the angle relative to the positive x-axis would be 90 - 130 = -40 degrees, and for the yacht, it would be 90 - 260 = -170 degrees.
Step 4: Convert the angles to radians.
Since trigonometric functions require angles in radians, convert the angles obtained in the previous step to radians. Multiply each angle by (π/180) to get the equivalent angle in radians. For the tugboat, the angle in radians would be (-40) * (π/180), and for the yacht, it would be (-170) * (π/180).
Step 5: Calculate the x and y components of each vector.
To find the x and y components of each vector, multiply the respective distance by the cosine of the angle and the sine of the angle, respectively.
For the tugboat, the x component would be 36 km * cos((-40) * (π/180)) and the y component would be 36 km * sin((-40) * (π/180)).
For the yacht, the x component would be 27 km * cos((-170) * (π/180)) and the y component would be 27 km * sin((-170) * (π/180)).
Step 6: Add the x and y components of the vectors.
To find the resultant vector from the ship to the yacht, add the x components and the y components obtained for each vector.
Step 7: Calculate the magnitude of the resultant vector.
To find the distance from the ship to the yacht, use the magnitude of the resultant vector obtained in the previous step. The magnitude can be calculated using the Pythagorean theorem: Magnitude = √(x^2 + y^2).
Step 8: Apply the scale to find the measured distance.
To find the measured distance, multiply the magnitude of the resultant vector by the scale factor of 1:450,000.
Finally, round the result to the nearest kilometer to get the answer.
If we label the craft S,T,Y respectively, then angle STY = 50°
So now just use the law of cosines to find SY
not sure what the scale has to do with it, since you ask for the actual distance in km ...