Three deer, A, B, and C, are grazing in a field. Deer B is located 61.1 m from deer A at an angle of 50.9° north of west. Deer C is located 78.4° north of east relative to deer A. The distance between deer B and C is 92.2 m. What is the distance between deer A and C?

To find the distance between deer A and C, we can use the Law of Cosines.

Let's call the distance between A and C as dAC.

Using the Law of Cosines, we have:

dAC^2 = dAB^2 + dBC^2 - 2 * dAB * dBC * cos(theta)

Where:
dAB is the distance between deer A and B (61.1 m)
dBC is the distance between deer B and C (92.2 m)
theta is the angle between dAB and dBC

First, let's find theta:

theta = 180° - angle A - angle C

angle A = 90° - 50.9° = 39.1° (angle of B north of west)
angle C = 180° - angle A - angle B
angle C = 180° - 39.1° - 78.4°
angle C = 62.5°

theta = 180° - 39.1° - 62.5°
theta = 78.4°

Now substituting the values into the Law of Cosines equation:

dAC^2 = (61.1)^2 + (92.2)^2 - 2 * 61.1 * 92.2 * cos(78.4°)

Calculating:
dAC^2 = 3732.21 + 8508.84 - 2 * 61.1 * 92.2 * cos(78.4°)

dAC^2 = 12240.05 - 10806.62

dAC^2 = 1433.43

Taking the square root of both sides to solve for dAC:

dAC = √1433.43

dAC ≈ 37.9 m

Therefore, the distance between deer A and C is approximately 37.9 meters.

To find the distance between deer A and C, we can use the law of cosines. The law of cosines is a formula used to find the length of a side of a triangle when you know the lengths of the other two sides and the angle between them.

Let's first visualize the situation:

We have deer A, deer B, and deer C forming a triangle.
Deer B is located 61.1 m away from deer A at an angle of 50.9° north of west.
Deer C is located 78.4° north of east relative to deer A.
The distance between deer B and C is 92.2 m.

Now, let's break it down step by step.

STEP 1: Calculate the distance between deer A and B.
We can use trigonometry to find the horizontal and vertical components of the distance between A and B.

Horizontal component:
cos(angle) = adjacent/hypotenuse
adjacent = cos(50.9°) x 61.1 m

Vertical component:
sin(angle) = opposite/hypotenuse
opposite = sin(50.9°) x 61.1 m

Now we can calculate the distance between A and B using the Pythagorean theorem:
distance_AB = √(horizontal_component^2 + vertical_component^2)

STEP 2: Calculate the angle between deer B and C.
Since the two deer are located relative to deer A, we need to find the angle between them.

angle_BAC = 180° - (angle_A + angle_C)

STEP 3: Apply the law of cosines to find the distance between A and C.
distance_AC = √(distance_AB^2 + distance_BC^2 - 2 x distance_AB x distance_BC x cos(angle_BAC))

Now you can plug in the values:

distance_AB = √(adjacent^2 + opposite^2)
distance_BC = 92.2 m
angle_BAC = 180° - (50.9° + 78.4°)

Finally, calculate the distance between deer A and C using the formula:

distance_AC = √(distance_AB^2 + distance_BC^2 - 2 x distance_AB x distance_BC x cos(angle_BAC))

This will give you the answer.