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.
To answer these questions, we can use the concepts of bearings and distance between points. Let's go step by step.
(a) The bearing of A and D:
First, let's visualize the situation. We have four points: A, B, C, and D. According to the information given, we know that B is 8 km directly East of A, C is 6 km from B on a bearing of 30 degrees, and D is 7 km from B on a bearing of 300 degrees.
We start by finding the bearing of C from A. Since C is 6 km from B at a bearing of 30 degrees, and B is 8 km East of A, we can use trigonometry to find the bearing of C from A.
Using the sine and cosine functions, we can calculate the sides of the triangle formed by A, B, and C. We have:
cosθ = adjacent/hypotenuse
sinθ = opposite/hypotenuse
Using the cosine function:
cosθ = AB/AC
cosθ = 8/AC
Solving for AC:
AC = 8/cosθ
Now we can find sinθ:
sinθ = BC/AC
sinθ = 6/(8/cosθ)
sinθ = 6cosθ/8
To find the bearing of C from A, we can use the inverse tangent function:
θ = arctan(sinθ/cosθ)
Substituting the values of sinθ and cosθ, we get:
θ = arctan((6cosθ/8)/cosθ)
By calculating this, you will find that the bearing of C from A is approximately 46.6 degrees.
Next, we need to find the bearing of D from B. Since D is 7 km from B on a bearing of 300 degrees, we can use the same trigonometric methods.
Using the sine and cosine functions, we can calculate the sides of the triangle formed by B, C, and D. We have:
sinη = opposite/hypotenuse
cosη = adjacent/hypotenuse
Using the sine function:
sinη = CD/BD
sinη = 7/BD
Solving for BD:
BD = 7/sinη
Now we can find cosη:
cosη = BC/BD
cosη = 6/(7/sinη)
cosη = 6sinη/7
To find the bearing of D from B, we can use the inverse tangent function:
η = arctan(sinη/cosη)
Substituting the values of sinη and cosη, we get:
η = arctan((7/sinη)/(6sinη/7))
By calculating this, you will find that the bearing of D from B is approximately 232.6 degrees.
Therefore, the bearing of A and D is the sum of the bearings of C from A (46.6 degrees) and D from B (232.6 degrees):
Bearing of A and D = 46.6 + 232.6 = 279.2 degrees
Therefore, the bearing of A and D is approximately 279.2 degrees.
(b) The distance BD in kilometers:
We already found the values of sinη and cosη in the previous calculation. The distance BD can be calculated using the sine function:
sinη = opposite/hypotenuse
sinη = 7/BD
Solving for BD:
BD = 7/sinη
By substituting the value of sinη, we can calculate BD:
BD = 7/(7/sinη) = 1/sinη
Therefore, BD is equal to 1/sinη.
(c) The perimeter of the ranch in kilometers:
To calculate the perimeter of the ranch, we need to add up the lengths of all the sides AB, BC, CD, and DA.
Given:
AB = 8 km (mentioned in the question)
BC = 6 km (mentioned in the question)
CD = 7 km (mentioned in the question)
DA = distance of point A from D
To find the distance of point A from D, we can use the law of cosines:
DA² = AC² + CD² - 2(AC)(CD)cos(180° - bearing of A and D)
Using the bearing of A and D we calculated earlier (279.2 degrees):
DA² = (8/cosθ)² + 7² - 2(8/cosθ)(7)cos(180° - 279.2°)
Solving this equation will give us the value of DA.
Now, to calculate the perimeter, we add up all the side lengths:
Perimeter = AB + BC + CD + DA
By substituting the values we have, we can calculate the perimeter of the ranch.
Remember to perform the necessary calculations to get the numerical answers for (b) and (c).