A bus driver driving south along a highway is 16 km from the nearest city. After traveling a distance of 32 km, the driver finds he is 19 km away from the same city. Find the bearing of the town from the second position of the motorist.
If we label the points order A,B,C, then we want the bearing of A from C.
Using the law of cosines, you could find the angles, and then compute the bearing.
Or, you can find that the intersection of
x^2+y^2 = 19^2
x^2 + (y+16)^2 = 32^2
is (14.1,12.7)
and now you can figure the bearing back to (0,0)
To find the bearing of the town from the second position of the motorist, we can use trigonometry.
First, let's draw a diagram to represent the situation.
```
City
|
|
(16 km)
|
Driver -----•-----
(32 km)
```
From the information given, we know that the driver drove a distance of 32 km from the first position and ended up 19 km away from the city. Let's call the distance between the second position of the driver and the city as "x" km.
Using trigonometry, we can create a right-angled triangle to solve for x. In this triangle, the hypotenuse is 32 km (distance traveled), one side is 19 km (distance from the city at the second position), and the other side is x km (distance between the second position and the city).
Using the Pythagorean theorem, we can write the equation:
x^2 + 19^2 = 32^2
Simplifying this equation:
x^2 + 361 = 1024
x^2 = 1024 - 361
x^2 = 663
To find the value of x, we take the square root of both sides:
x = sqrt(663)
Now we have the value of x, which is approximately 25.75 km.
To find the bearing of the town from the second position of the motorist, we need to determine the angle between the direction from the second position to the city and the north direction.
To find this angle, we can use trigonometry again. We know that one side of the triangle (opposite to the angle) is 19 km, and the other side (adjacent to the angle) is 25.75 km.
Using the tangent function:
tan(angle) = opposite / adjacent
tan(angle) = 19 / 25.75
Solving for the angle:
angle = atan(19 / 25.75)
The arctangent (atan) function gives us the angle in radians. To convert it to degrees, we can multiply by 180 and divide by π (pi).
angle_degrees = atan(19 / 25.75) * (180 / π)
Thus, the bearing of the town from the second position of the motorist is approximately equal to the calculated angle_degrees.