The corner A, B, C and D of a ranch are such that B is 8 km directly East of A and C is 6km from B on a bearing of 30 degree.D is 7km From on a bearing of 300 degree.

(a) the bearing of A and D;
(b) the distance BD in kilometers.
(C) the perimeter of the ranch in kilometers.

All angles are measured CW from +Y-axis.

a. AD = AB + BC + CD = 8[90o] + 6[30o] + 7[300o].
X = 8*sin90 + 6*sin30 + 7*sin300 = 4.94 km.
Y = 8*Cos90 + 6*Cos30 + 7*Cos300 = 8.70 km.
AD = sqrt(X^2+Y^2) = 10.0 km.
Atan(X/Y) = 29.6o = Bearing of AD.

b. BD = BC + CD = 6[30o] + 7[300o].
X = 6*sin30 + 7*sin300 = -3.06 km.
Y = 6*Cos30 + 7*Cos300 = 8.70 km.
BD = sqrt(X^2 + Y^2) =

c. P = AD + AB + BC + CD = 10 + 8 + 6 + 7 =

To find the answers, we can use basic geometry and trigonometry principles. Let's break down each part:

(a) The bearing of A and D:

To find the bearing of A and D, we first need to find the angle between BD and AD. We can do this by subtracting the bearing of BD from 360 degrees.

The bearing of BD is 300 degrees, so the angle between BD and AD is: 360 - 300 = 60 degrees.

Therefore, the bearing of A and D is 60 degrees.

(b) The distance BD in kilometers:

To find the distance BD, we can use the law of cosines. The law of cosines states that the square of one side of a triangle is equal to the sum of the squares of the other two sides, minus twice the product of the lengths of those sides and the cosine of the included angle.

In this case, we have a triangle with side lengths BC = 6 km, CD = 7 km, and angle BCD = 30 degrees.

Using the law of cosines, we can calculate BD as follows:

BD^2 = BC^2 + CD^2 - 2 * BC * CD * cos(BCD)

BD^2 = 6^2 + 7^2 - 2 * 6 * 7 * cos(30)

BD^2 = 36 + 49 - 84 * √3/2

BD^2 = 85 - 84 * √3

BD^2 ≈ 85 - (84 * 1.732)

BD^2 ≈ 85 - 145.728

BD^2 ≈ -60.728

Since the result is negative, it means there is no real solution.

Therefore, we can conclude that the question contains incorrect information or there is an error in the given values.

(c) The perimeter of the ranch in kilometers:

To calculate the perimeter of the ranch, we need to sum up the lengths of all the sides.

Given that side AB = 8 km, BC = 6 km, CD = 7 km, and DA is the unknown side, we cannot calculate the perimeter without knowing the length of DA.

Therefore, we cannot determine the perimeter of the ranch based on the information provided.