Town P is on bearing 315 degree from town Q. While town R is south of town P and west of town Q. If town R is 60km away from Q, how fara is R from P?
To determine the distance between town P and town R, we can use the concept of right-angled triangles and trigonometry.
First, let's imagine an imaginary line connecting town P and town Q. Since town P is on a bearing of 315 degrees from town Q, this means that the line connecting them would make a 45-degree angle with the vertical.
Since town R is south of town P and west of town Q, town R lies on the imaginary line connecting P and Q. Considering that the distance between town Q and town R is 60 km and that town R lies south of town P, we can conclude that town R is 60 km away from Q.
Now, we have created a right-angled triangle with a 45-degree angle and a hypotenuse of 60 km. We need to find the length of the side connecting town P and town R. To do this, we can use the sine and cosine functions.
In a triangle, the sine of an angle is equal to the length of the side opposite the angle divided by the length of the hypotenuse. In this case, the length of the side opposite the 45-degree angle is the distance between P and R, which we'll denote as x. The length of the hypotenuse is 60 km.
So, we can set up the following equation:
sin(45 degrees) = x/60 km
Using the value of sin(45 degrees) = 1/sqrt(2), we can solve for x:
x/60 km = 1/sqrt(2)
Multiplying both sides of the equation by 60 km:
x = 60 km / sqrt(2)
Simplifying the expression by multiplying the numerator and denominator by sqrt(2):
x = 60 km * sqrt(2) / (sqrt(2) * sqrt(2))
x = 60 km * sqrt(2) / 2
x = 30 km * sqrt(2)
Therefore, the distance between town P and town R is approximately 30 km * sqrt(2).
To determine the distance between town P and town R, we can break down this problem into simpler steps.
Step 1: Determine the compass directions of town R in relation to town P and town Q:
- Town P is located on a bearing of 315 degrees from town Q. This means that an imaginary line drawn from town Q to town P would form an angle of 45 degrees counter-clockwise (360 - 315) from the eastward direction.
- Town R is south of town P and west of town Q. South is a direction opposite to the north, and west is a direction 90 degrees counter-clockwise from the north.
Step 2: Combine the directions to determine the bearing from town R to town P:
- Since town R is south of town P, we can add 180 degrees to the bearing from town Q to town P (which is 45 degrees counter-clockwise from the eastward direction).
- 45 degrees + 180 degrees = 225 degrees.
- Therefore, the bearing from town R to town P is 225 degrees.
Step 3: Apply trigonometry to solve for the distance between town P and town R:
- Given that town R is 60km away from town Q, we need to determine the distance between town P and town R.
To calculate that distance, we can use the Law of Cosines, which states:
c^2 = a^2 + b^2 - 2ab * cos(C)
In this case:
- 'c' is the distance between town P and town R (the side opposite to angle C).
- 'a' is the distance between town Q and town R.
- 'b' is the distance between town P and town Q.
- Angle C is 225 degrees.
Since we don't have the value for 'a', we can use the Law of Sines to relate angle C to the respective sides:
a / sin(A) = c / sin(C)
We know 'c' (60km) and angle C (225 degrees), and we can consider angle A (which represents the angle between sides 'a' and 'b') as 45 degrees (since the sum of angles in a triangle is 180 degrees).
Now we can solve for 'a' using the Law of Sines:
a = (c * sin(A)) / sin(C)
= (60 * sin(45)) / sin(225)
Using a calculator, we can compute the value of a to be approximately 42.43 km.
Therefore, town R is approximately 42.43 km away from town P.