from the top of a vertical mast 140m high. two lorries on the same ground lavel 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 correct to 3 s.f.
I need the diagram and the answer as soon as possible
draw a diagram and review the basic trig functions. It should be clear that the distance
x = 140 cot38° + 140 cot65°
To solve this problem, we can use trigonometry and the concept of angles of depression. Let's break it down step by step:
Step 1: Draw a diagram
Draw a diagram to visualize the situation. Label the vertical mast as "M," the lorry on the east as "L1," the lorry on the west as "L2," and mark the vertical height of the mast as 140m.
L1 ---------------------- M ---------------------- L2
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140m
Step 2: Calculate the height difference between the mast and the lorries
Since the mast is vertical and the lorries are on the same ground level, the height difference between the mast and each lorry is the same and equal to the vertical height of the mast, which is 140m.
Step 3: Identify the right-angled triangles
In the diagram, we have two right-angled triangles: ΔLM1A and ΔLM2B, where A and B are the positions directly beneath L1 and L2, respectively.
L1 ---------------------- M ---------------------- L2
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140m
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A
Step 4: Calculate the lengths of LM1 and LM2
Using the angles of depression given, we can calculate the lengths of LM1 and LM2 using the tangent ratio.
Let's start with LM1:
tan(65°) = LM1 / 140
LM1 = tan(65°) * 140
Now for LM2:
tan(38°) = LM2 / 140
LM2 = tan(38°) * 140
Step 5: Calculate the distance between the two lorries
To find the distance between L1 and L2, we need to determine the horizontal distance between A and B. This distance is equal to the difference between LM1 and LM2.
Distance between L1 and L2 = LM1 - LM2
Step 6: Round the answer to 3 significant figures
After performing the calculation in Step 5, round the final answer to 3 significant figures to maintain consistency with the given data.
So the final answer will be the distance between the two lorries, rounded to 3 significant figures.
To solve this problem, we will use trigonometry and the concept of angles of depression.
Let's denote:
- The distance between the mast and the lorry to the east as "d1."
- The distance between the mast and the lorry to the west as "d2."
First, we can determine the height of the mast using the angle of depression of 65°. By drawing a right triangle, the height of the mast will be the opposite side, and the distance from the mast to the lorry to the east will be the adjacent side.
Since we know the angle (65°) and the height of the mast (140m), we can use the trigonometric function tangent:
tan(65°) = height of the mast (140m) / d1
Rearranging the equation, we get:
d1 = height of the mast (140m) / tan(65°)
Next, by using the angle of depression of 38° for the lorry to the west, we form another right triangle. This time, the height of the mast will still be the opposite side, and the distance from the mast to the lorry to the west will be the adjacent side.
Using the same logic as before, we can set up the following equation:
tan(38°) = height of the mast (140m) / d2
Rearranging the equation, we get:
d2 = height of the mast (140m) / tan(38°)
Now, to find the distance between the two lorries, we can subtract d1 from d2:
distance between the lorries = d2 - d1
Substituting the equations we found earlier, we have:
distance between the lorries = (height of the mast (140m) / tan(38°)) - (height of the mast (140m) / tan(65°))
Now, let's plug in the values and calculate the distance:
distance between the lorries = (140m / tan(38°)) - (140m / tan(65°))
Using a scientific calculator, we find that the distance between the lorries is approximately 317.532 meters (to 3 significant figures).
Therefore, the distance between the two lorries, correct to 3 significant figures, is 317.532 meters.