A bee flits from flower to flower. If it leaves the hive and travels 210 m [N32°E] and then 37 m [S51°W], how far and in what direction is the bee from its hive?
North= 210 cos 32 - 37 cos 51
East = 210 sin 32 - 37 sin 51
distance = sqrt (North^2 + East^2)
tan angle east of north = East/North
Using standard math-trig notation:
N32°E ----> 58°
S51°W ---> 219°
resultant = 210(cos58, sin58) + 37(cos219,sin219)
= (111.283, 178.090) + (-28.7544, -23.285)
= (82.529, 154.805)
Magnitude = √(82.283^2 + 154.805^2) = 175.43 m
for angle:
tan (angle) = 154.805/82.529 = 1.8758
angle = 61.9°
or N28.1°E in your notation.
To find the distance and direction of the bee from its hive, we can use vector addition.
Step 1: Convert the given distances and directions into component form.
The distance traveled north can be decomposed into its north and east components using trigonometry:
Distance north = 210 m * sin(32°) ≈ 110.34 m
Distance east = 210 m * cos(32°) ≈ 176.88 m
The distance traveled south can be decomposed into its south and west components:
Distance south = 37 m * sin(51°) ≈ 29.80 m
Distance west = 37 m * cos(51°) ≈ 23.43 m
Step 2: Add the components of the distances traveled.
North component = Distance north - Distance south = 110.34 m - 29.80 m = 80.54 m (North)
East component = Distance east - Distance west = 176.88 m - 23.43 m = 153.45 m (East)
Step 3: Use the Pythagorean theorem to find the total distance from the hive.
Distance = √(North component^2 + East component^2)
Distance = √(80.54^2 + 153.45^2)
Distance ≈ 172.42 m
Step 4: Use trigonometry to find the direction from the hive.
Direction = arctan(East component / North component)
Direction = arctan(153.45 m / 80.54 m)
Direction ≈ 61.88°
Therefore, the bee is approximately 172.42 m away from its hive in a direction of N61°E.
To find the distance and direction from the hive, we can use vector addition.
First, let's break down the bee's movements into two components: north-south (N/S) and east-west (E/W).
We have:
- 210 m [N32°E] - This means the bee traveled 210 m in a direction 32° east of north.
- 37 m [S51°W] - This means the bee traveled 37 m in a direction 51° west of south.
To find the N/S component, we need to calculate the vertical distance traveled. We can do this by using the sine function:
N/S component = Total distance * sin(theta)
For the E/W component, we'll use the cosine function:
E/W component = Total distance * cos(theta)
Now let's calculate the N/S and E/W components separately.
N/S component:
N/S component = 210 m * sin(32°)
≈ 111.347 m (rounded to 3 decimal places)
E/W component:
E/W component = 37 m * cos(51°)
≈ 23.467 m (rounded to 3 decimal places)
To find the total distance from the hive, we can use the Pythagorean theorem:
Total distance = √(N/S component^2 + E/W component^2)
Total distance = √(111.347^2 + 23.467^2)
≈ √12396.709
≈ 111.258 m (rounded to 3 decimal places)
Finally, to determine the direction, we can calculate the angle between the resultant vector and the north direction:
Angle = arctan(E/W component / N/S component)
Angle = arctan(23.467 / 111.347)
≈ 12.003 degrees (rounded to 3 decimal places)
Therefore, the bee is approximately 111.258 meters away from the hive in a direction of N12°E.