A bushwalking party leave P and walk on a bearing of 335(degrees) for 11.4km until they reach Q. From Q they walk on a bearing of 65(degrees) for 18.7km at which point they arrive at R.
a) What is the distance between R and P to 3 sig. figs.?
b) What is the bearing of R from P, correct to nearest minute.
one walks on a heading, not a bearing.
a) Note that angle Q is 90 degrees, so you have a right triangle. Easy...
b) This is the correct use of "bearing." If the bearing is x, then
tan(x+25) = 18.7/11.4
A diagram will show you why.
q is 130 deg for me o-o, but i prob did something wrong
To find the distance between R and P, you can use the concept of vectors or the Pythagorean theorem. Let's use the Pythagorean theorem in this case.
a) To find the distance between R and P, we need to find the horizontal and vertical components of the triangle formed by the points P, Q, and R.
First, let's start by finding the horizontal distance between P and Q. The bearing of 335 degrees indicates a direction that is 55 degrees clockwise from the positive x-axis (east). We can represent this as a vector (x, y), where x represents the horizontal displacement and y represents the vertical displacement.
To find the horizontal distance, we use the formula: x = distance * cos(bearing)
x = 11.4 km * cos(55 degrees) ≈ 6.618 km
Next, let's find the horizontal distance between Q and R. The bearing of 65 degrees indicates a direction that is 65 degrees clockwise from the positive x-axis (east).
x = 18.7 km * cos(65 degrees) ≈ 7.898 km
To find the total horizontal distance between P and R, we add the horizontal distances between P and Q, and between Q and R:
Total horizontal distance = 6.618 km + 7.898 km = 14.516 km
Next, let's find the vertical distance between P and Q. The bearing of 335 degrees indicates a direction that is 55 degrees above the positive x-axis (east).
To find the vertical distance, we use the formula: y = distance * sin(bearing)
y = 11.4 km * sin(55 degrees) ≈ 9.410 km
Next, let's find the vertical distance between Q and R. The bearing of 65 degrees indicates a direction that is 25 degrees above the positive x-axis (east).
y = 18.7 km * sin(25 degrees) ≈ 7.945 km
To find the total vertical distance between P and R, we add the vertical distances between P and Q, and between Q and R:
Total vertical distance = 9.410 km + 7.945 km = 17.355 km
Now, we can use the Pythagorean theorem to find the distance between P and R:
Distance = √(horizontal distance^2 + vertical distance^2)
Distance = √(14.516 km^2 + 17.355 km^2) = √(210.971256 km^2) ≈ 14.523 km
Therefore, the distance between R and P is approximately 14.523 km to 3 significant figures.
b) To find the bearing of R from P, we can use trigonometry.
The bearing can be found by calculating the angle of the line PR with respect to the positive x-axis (east).
We can use the formula: bearing = arctan(vertical distance / horizontal distance)
Using the previously calculated values:
bearing = arctan(17.355 km / 14.516 km)
Using a calculator, the approximate value of the bearing is 49.25 degrees.
To find the nearest minute, round the decimal part:
49.25 degrees ≈ 49 degrees
Therefore, the bearing of R from P is approximately 49 degrees.