a plane starts from a point P and flies 240km on a bearing of 120degrees to Q. At Q the plane makes a 90degree turn and flies back 100km on a bearing of 210degrees to R. At R the plane again changes direction and flies back to P.
The direct distance, in km, from R to P is?
the 3 figure bearing the pilot uses to fly from R to P is?
To find the direct distance from R to P, we can use the Pythagorean theorem. The plane originally flew 240km on a bearing of 120 degrees from P to Q. Then, it flew back 100km on a bearing of 210 degrees from Q to R. Now, let's calculate the remaining distance from R to P.
The remaining distance is the hypotenuse of the right triangle formed by the triangle PQR. The sides of this triangle are: PQ = 240km and QR = 100km.
Using the Pythagorean theorem, we can calculate the direct distance RP:
RP^2 = PQ^2 + QR^2
RP^2 = (240km)^2 + (100km)^2
RP^2 = 57600km^2 + 10000km^2
RP^2 = 67600km^2
Taking the square root of both sides, we get:
RP ≈ √67600km^2
RP ≈ 260km
Therefore, the direct distance from R to P is approximately 260km.
To determine the bearing the pilot uses to fly from R to P, we need to calculate the angle formed by the triangle PQR at R. We can use the Law of Cosines to find this angle.
In triangle PQR, the side lengths are: PQ = 240km, QR = 100km, and RP = 260km (which we just calculated). Using the Law of Cosines:
RP^2 = PQ^2 + QR^2 - 2 * PQ * QR * cos(angle R)
260km^2 = 240km^2 + 100km^2 - 2 * 240km * 100km * cos(angle R)
Simplifying further:
67600km^2 = 57600km^2 + 10000km^2 - 48000km * cos(angle R)
Rearranging the equation:
48000km * cos(angle R) = 20000km^2
cos(angle R) = 20000km^2 / 48000km
cos(angle R) ≈ 0.4166
Now we can find the angle R using the inverse cosine function:
angle R ≈ arccos(0.4166)
angle R ≈ 65.996 degrees
Therefore, the 3-figure bearing the pilot uses to fly from R to P is approximately 066 degrees.
To find the direct distance from R to P, we need to use the concept of the Pythagorean theorem.
We know that the plane traveled 240km from P to Q and then 100km from Q to R. Thus, the distance from R to P can be found by calculating the hypotenuse of a right-angled triangle with sides of 240km and 100km.
Using the Pythagorean theorem:
Direct distance from R to P = √(240^2 + 100^2) = √(57600 + 10000) = √(67600) ≈ 260.77 km
Therefore, the direct distance from R to P is approximately 260.77 km.
To find the bearing the pilot uses to fly from R to P, we can use the inverse tangent function to calculate the angle.
Using the inverse tangent of the ratio of the opposite and adjacent sides of a right triangle:
Angle = arctan(opposite/adjacent) = arctan(240/100)
Angle = arctan(2.4) ≈ 68.2 degrees
Since the bearing is measured clockwise from the north, we need to subtract the angle from 360 degrees:
Bearing from R to P = 360° - 68.2° ≈ 291.8 degrees
Therefore, the 3-figure bearing the pilot uses to fly from R to P is approximately 292 degrees.
to find PR, use the Pythagorean Theorem, since angle Q is a right angle.
Knowing PR, you can use the law of sines to find angle R.
If you've drawn your diagram carefully, the bearing is 360-((angle R)-30)