Question

A worker bee flies 504m south from a rose garden to the hive; she gets the message that there are many more flowers 993m southeast. How far is it from the first Flower field to the second?

Answer 1

Answer on Question #38857, Physics, Mechanics | Kinematics | Dynamics

A worker bee flies 504m south from a rose garden to the hive; she gets the message that there are many more flowers 993m southeast. How far is it from the first flower field to the second?

Solution.

We introduce the following notation.

a = 504 m (distance from a rose garden to the hive),

$b = 993 \, \text{m}$ (distance from a hive to the second flower field),

c - ? (distance from the first flower field to the second flower field),

$\gamma = 135{}^{\circ}$ (angle between the direction to the south and south-east direction).

I give the formula for the Law of Cosines and use it to find the missing side length of a triangle.

$$c ^ {2} = a ^ {2} + b ^ {2} - 2 a b \cos \gamma$$$$c ^ {2} = 5 0 4 ^ {2} + 9 9 3 ^ {2} - 2 \cdot 5 0 4 \cdot 9 9 3 \cdot \cos (1 3 5 {}^ {\circ})$$ $$\cos (1 3 5 {}^ {\circ}) = - 0. 7 0 7 1$$ $$c ^ {2} = 1 9 4 7 8 3 2. 5$$ $$c = \sqrt {1 9 4 7 8 3 2 . 5} = 1 3 9 5. 6 5 \approx 1 3 9 6 \mathrm {m}.$$

Answer. 1396 m.