From the top of a vertical mast 140m high,two lorries on the same ground level are observed, one due east and the other due west of the mast.If their angles of depression are 65 and 38 respectively; calculate the distance between the two lorries
tan 65 = 140/d1
tan 38 = 140/d2
total distance = d1 + d2
what is the answer
To calculate the distance between the two lorries, we can use trigonometric functions and the given angles of depression.
Let's assume the distance between the mast and the eastern lorry is x, and the distance between the mast and the western lorry is y.
From the top of the mast, we can draw two right triangles, one for each lorry. The angle at the top of both triangles is 90 degrees.
For the eastern lorry:
The angle of depression is 65 degrees, and the height of the mast is 140m. So, in this triangle, we have:
tan(65) = height of the mast / distance to the eastern lorry
tan(65) = 140 / x
For the western lorry:
The angle of depression is 38 degrees, and the height of the mast is 140m. So, in this triangle, we have:
tan(38) = height of the mast / distance to the western lorry
tan(38) = 140 / y
Now, we need to solve these two equations to find the values of x and y. Then, we can calculate the distance between the two lorries.
First, let's solve the equation with tan(65):
tan(65) = 140 / x
To find x, we can rearrange the equation:
x = 140 / tan(65)
Using a calculator, we find that x ≈ 70.27m
Next, let's solve the equation with tan(38):
tan(38) = 140 / y
Again, rearranging the equation to solve for y:
y = 140 / tan(38)
Using a calculator, we find that y ≈ 157.73m
Now, we can calculate the distance between the two lorries by subtracting the distances from the mast to each lorry:
Distance between the two lorries = y - x
≈ 157.73m - 70.27m
≈ 87.46m
So, the distance between the two lorries is approximately 87.46 meters.
To solve this problem, we can start by drawing a diagram to visualize the given information. Let's label the points as follows:
- A: Top of the vertical mast
- B: Lorry in the east
- C: Lorry in the west
Next, let's calculate the distances AB and AC using trigonometry.
1. Angle of depression at point B (east lorry) = 65 degrees.
This means that angle ABC (the angle of elevation from the east lorry to the top of the mast) is also 65 degrees. Since the angle at the top of the mast is a right angle (90 degrees), we can conclude that angle BAC is 90 - 65 = 25 degrees.
2. Angle of depression at point C (west lorry) = 38 degrees.
This means that angle ACB (the angle of elevation from the west lorry to the top of the mast) is also 38 degrees. Since the angle at the top of the mast is a right angle (90 degrees), we can conclude that angle BCA is 90 - 38 = 52 degrees.
Now, we can use trigonometry to calculate the distances AB and AC.
1. Distance AB:
We know the height of the mast (140m) and the angle BAC (25 degrees). Since we have the opposite and adjacent sides, we can use the tangent function:
tan(25) = AB / 140
AB = tan(25) * 140
2. Distance AC:
We know the height of the mast (140m) and the angle BCA (52 degrees). Since we have the opposite and adjacent sides, we can use the tangent function:
tan(52) = AC / 140
AC = tan(52) * 140
Finally, we can calculate the distance between the two lorries by subtracting the distances AB and AC:
Distance between the two lorries = AB - AC
By plugging in the values and calculating the equations, you can find the exact distance between the two lorries.