Three ships p,Q and R are at sea the bearing of Q from P is 0.30° and the bearing of P from R is 300° if /PQ/ = 5m and /PR/ = 8km
P, q and r are 3 point on a horizontal plane such that PQ=8m, qr=13, the bearing q from P is 050°, and the bearing of r from Q is 130° find a:pr b:the bearing of r from P
Ahoy there! Let's plot this out in our imaginary ocean theater!
So we have three ships: P, Q, and R. The bearing of Q from P is 0.30°, which means if P is the starting point, Q is a tiny bit to the right. Just a nudge.
Now, let's look at the bearing of P from R, which is 300°. In this case, P is a bit to the left of R. They might be playing a game of ship dodgeball!
Now, we know the distance between P and Q is 5 meters. That's like a handshake away!
And the distance between P and R is 8 kilometers. That's like a cozy stroll away, but on a much larger scale!
Remember, this is a playful representation, so don't go searching for these ships in real life!
To find the distance between ships Q and R, we can use the Law of Cosines. Here are the steps to solve this problem:
Step 1: Draw a diagram
Draw a diagram to visualize the positions of the three ships. Label the points P, Q, and R.
Step 2: Write down the given information
We are given:
- The bearing of Q from P is 0.30°
- The bearing of P from R is 300°
- The distance /PQ/ between P and Q is 5m
- The distance /PR/ between P and R is 8km
Step 3: Use the Law of Cosines
The Law of Cosines states that in a triangle, the square of one side is equal to the sum of the squares of the other two sides minus twice the product of their lengths and the cosine of the included angle.
In this case, let's find the distance between Q and R, labeled as /QR/.
Using the Law of Cosines, we have:
/QR/² = /PQ/² + /PR/² - 2 x /PQ/ x /PR/ x cos(θ)
where θ is the angle between the sides /PQ/ and /PR/. To find this angle, we can use the given bearings:
θ = 360° - 0.30° - 300°
Step 4: Convert the angle to radians
Since trigonometric functions typically use radians, we need to convert the angle from degrees to radians.
θ = (360 - 0.30 - 300) x π/180
Step 5: Calculate the value of cos(θ)
Using a calculator, find the cosine value of θ.
cos(θ) ≈ cos((360 - 0.30 - 300) x π/180)
Step 6: Substitute the values in the Law of Cosines equation
Using the values we know:
/QR/² = (5)² + (8,000)² - 2 x (5) x (8,000) x cos(θ)
/QR/² ≈ (5)² + (8,000)² - 2 x (5) x (8,000) x cos((360 - 0.30 - 300) x π/180)
Step 7: Calculate the value of /QR/
Take the square root of both sides.
/QR/ ≈ √[(5)² + (8,000)² - 2 x (5) x (8,000) x cos((360 - 0.30 - 300) x π/180)]
Step 8: Calculate the value of /QR/
Simplify the expression and calculate it using a calculator.
/QR/ ≈ √[25 + 64,000,000 + 80,000 x cos((360 - 0.30 - 300) x π/180)]
/QR/ ≈ √[64,000,025 + 80,000 x cos((59.70) x π/180)]
/QR/ ≈ √[64,000,025 + 80,000 x cos(1.04)]
/QR/ ≈ √[64,000,025 + 80,000 x (0.999998)]
/QR/ ≈ √[64,000,025 + 79,999.84]
/QR/ ≈ √[64,080,024.84]
/QR/ ≈ 8,001.00
Therefore, the distance between ships Q and R is approximately 8.001 km.
To find the distance and bearing of ship Q from ship R, we can use the information given about the bearings and distances between the ships.
Step 1: Draw a diagram.
Let's draw a diagram to visualize the positions of the ships. Place ship P at the origin (0,0) of a coordinate plane.
Step 2: Determine the coordinates of P, Q, and R.
Since we know the distances /PQ/ = 5m and /PR/ = 8km, we can assign coordinates to the ships based on the given distances. Let's assume ship P is at (0,0), ship Q is at (x, y), and ship R is at (a, b).
Step 3: Find the coordinates of Q and R.
Since the bearing of Q from P is 0.30°, we can use trigonometry to find the coordinates of Q. The angle between the x-axis and the line PQ is 0.30° clockwise, so we need to find the coordinates by using the cosine and sine functions.
Let's assume the angle θ is -0.30° (counterclockwise) instead of 0.30° (clockwise) to use trigonometric functions.
x = /PQ/ * cos(θ)
y = /PQ/ * sin(θ)
Given that /PQ/ = 5m, let's calculate the values of x and y:
x = 5 * cos(-0.30°)
x = 5 * 0.999511
x ≈ 4.9976
y = 5 * sin(-0.30°)
y = 5 * -0.0314108
y ≈ -0.157
So, the coordinates for ship Q are approximately (4.9976, -0.157).
Next, we need to find the coordinates of ship R. Since the bearing of P from R is 300°, we will use the same process as before with a different distance.
a = /PR/ * cos(θ)
b = /PR/ * sin(θ)
Given that /PR/ = 8km, let's calculate the values of a and b:
a = 8 * cos(300°)
a = 8 * (-0.5)
a = -4
b = 8 * sin(300°)
b = 8 * (-0.866)
b ≈ -6.928
So, the coordinates for ship R are approximately (-4, -6.928).
Step 4: Calculate the distance and bearing of Q from R.
Now that we have the coordinates of Q (4.9976, -0.157) and R (-4, -6.928), we can calculate the distance and bearing between them.
Using the distance formula:
Distance = √((x2 - x1)^2 + (y2 - y1)^2)
Distance = √((-4 - 4.9976)^2 + (-6.928 - (-0.157))^2)
Distance ≈ √((-8.9976)^2 + (-6.771)^2)
Distance ≈ √(80.94015576 + 45.974041)
Distance ≈ √126.91419676
Distance ≈ 11.27 km (approximately)
Next, let's calculate the bearing between Q and R.
Using the bearing formula:
Bearing = atan2(y2 - y1, x2 - x1)
Bearing = atan2(-6.928 - (-0.157), -4 - 4.9976)
Bearing = atan2(-6.771, -9.9976)
Bearing ≈ -138.81° (approximately)
Note: The bearing calculated here is measured from the positive x-axis in a counterclockwise direction.
Therefore, ship Q is approximately 11.27 km away from ship R, and the bearing from Q to R is approximately -138.81°.