There are two sign posts A and B on the edge of the road. A is 400 m to the west of b. A tree is on a bearing of 60 degree from A and a bearing of 330 degree from B Calculate the shortest distance of the tree from the edge of the road.
Draw a diagram. You have a 30-60-90 triangle with hypotenuse 400.
That should help.
To calculate the shortest distance of the tree from the edge of the road, we can use the concept of trigonometry and vectors. Here's how you can calculate it step by step:
1. Draw a diagram: Start by drawing a diagram to visualize the problem. Draw two signposts, A and B, on the edge of the road. Place the tree on a bearing of 60 degrees from point A and a bearing of 330 degrees from point B. Label the distance between A and B as 400 meters.
2. Calculate the position of the tree: Since the tree is on a bearing of 60 degrees from point A, we can calculate its position relative to A using trigonometry. Let's call the distance from point A to the tree "x meters." In a right-angled triangle with angle 60 degrees, the adjacent side is x meters and the hypotenuse is the shortest distance of the tree from the edge of the road. Therefore, we need to find the adjacent side length.
Using trigonometry, we can say that:
cos(60 degrees) = adjacent side (x meters) / hypotenuse
cos(60 degrees) = x / hypotenuse
x = cos(60 degrees) * hypotenuse
3. Find the hypotenuse: To find the hypotenuse, we can use the distance between A and B, which is given as 400 meters. Since the tree is 400 meters to the west of B, we can consider the line connecting B to the tree as the hypotenuse of a right-angled triangle. This means the hypotenuse is already given as 400 meters.
4. Calculate x: Using the equation from step 2, we can calculate the value of x by substituting the known values:
x = cos(60 degrees) * 400 meters
Now, let's calculate the value of x:
cos(60 degrees) = 0.5 (approximately)
x = 0.5 * 400 meters
x = 200 meters
Therefore, the distance from point A to the tree is 200 meters.
5. Calculate the total shortest distance: To calculate the shortest distance of the tree from the edge of the road, we need to find the perpendicular distance from point A to the line connecting A and B. This can be calculated using the Pythagorean theorem.
Let's call the perpendicular distance from point A to the road "y meters." In a right-angled triangle with side lengths x and y, and hypotenuse 400 meters, we can say:
y^2 + x^2 = 400^2
Since we know the value of x from step 4 (x = 200 meters), we can substitute it in the equation:
y^2 + (200 meters)^2 = 400^2
y^2 + 40,000 meters^2 = 160,000 meters^2
y^2 = 120,000 meters^2
y ≈ 346.41 meters (approximately)
Therefore, the shortest distance of the tree from the edge of the road is approximately 346.41 meters.