1. A dog sits under the only shade tree in its territory. Suddenly the dog desires to dig a previously buried bone. From the tree the dog goes 13 meters at 22.6° north of east. Not finding the bone there the dog then goes 17 meters at 61.9° north of west. Digging at the second location yields a bone. To go directly from the tree to the bone the dog should have gone what distance in what direction? Answer: 20.4m @ 78.7° N of E

Question

1. A dog sits under the only shade tree in its territory. Suddenly the dog desires to dig a previously buried bone. From the tree the dog goes 13 meters at 22.6° north of east. Not finding the bone there the dog then goes 17 meters at 61.9° north of west. Digging at the second location yields a bone. To go directly from the tree to the bone the dog should have gone what distance in what direction? Answer: 20.4m @ 78.7° N of E

The answer is given above, but I am confused as to how to get to the answer using the law of cosine and sine. Thanks!

Answer 1

Disp. = 13m[22.6o] + 17[118.1o].

Disp. = (13*Cos22.6+17*Cos118.1) + (13*sin22.6+17*sin118.1)i,
Disp. = 4 + 20i = 20.4m[78.7o N. of E].

Answer 2

To solve this problem using the law of cosines and sines, we can break down the dog's movements into two separate vectors: one from the tree to the first location, and another from the first location to the second location.

Let's define the following variables:
- Distance from the tree to the first location: d1
- Angle between the tree and the first location: θ1 (22.6° north of east)
- Distance from the first location to the second location: d2
- Angle between the first location and the second location: θ2 (61.9° north of west)

Now, let's calculate the dog's displacement from the tree to the second location using the law of cosines:

1. Calculate the internal angle between the two vectors:
α = 180° - (θ1 + θ2)

2. Use the law of cosines to find the displacement:
displacement^2 = d1^2 + d2^2 - 2 * d1 * d2 * cos(α)

3. Take the square root of the result to get the displacement from the tree to the second location.

Next, we need to find the angle at which the dog should have gone directly from the tree to the bone. We can use the law of sines for this:

1. Use the law of sines to find the angle opposite to d1:
sin(θ1) / displacement = sin(α) / d2

2. Take the inverse sine of both sides to find θ1.

Now we have the displacement and θ1. We can now determine the distance and direction the dog should have gone to go directly from the tree to the bone:

1. The distance is equal to the displacement from the tree to the second location.

2. The direction is equal to the angle opposite to the displacement.

By following these steps, you should be able to obtain the answer of 20.4m at 78.7° north of east.