An observer on a cliff 1000 dm above sea level sights two ships due east. the angles of depression of the ships are 44o and 32o. Find the distance between the ships.
draw a diagram. If the near ship is at distance x from shore, and the ships are d apart, then
x = 1000 cot44°
x+d = 1000 cot32°
so, d = 1000(cot32°- cot44°)
To find the distance between the ships, we can use trigonometry. Let's call the distance between the observer and the first ship as 'x', and the distance between the observer and the second ship as 'y'. We'll calculate both 'x' and 'y', and then find the difference between them to find the distance between the ships.
Using the angle of depression of 44° for the first ship, we can form the equation:
tan(44°) = x / 1000 dm
Rearranging this equation, we get:
x = 1000 dm * tan(44°)
Using a calculator, we can calculate the value of x:
x ≈ 1000 dm * 0.9654
x ≈ 965.4 dm
Using the angle of depression of 32° for the second ship, we can form the equation:
tan(32°) = y / 1000 dm
Rearranging this equation, we get:
y = 1000 dm * tan(32°)
Using a calculator, we can calculate the value of y:
y ≈ 1000 dm * 0.6249
y ≈ 624.9 dm
Now, to find the distance between the ships, we subtract the two distances:
Distance between the ships = y - x
Distance between the ships ≈ 624.9 dm - 965.4 dm
Distance between the ships ≈ -340.5 dm
Since the result is negative, it means that the second ship is closer to the observer than the first ship. To find the positive distance, we take the absolute value:
Distance between the ships ≈ | -340.5 dm |
Distance between the ships ≈ 340.5 dm
Therefore, the distance between the ships is approximately 340.5 dm.
To find the distance between the ships, we can use trigonometry and the concept of angles of depression. Let's break down the problem step by step:
Step 1: Draw a diagram.
Sketch a diagram that represents the situation described in the problem. This will help visualize the given information and guide us through the solution process.
Cliff (1000 dm above sea level)
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Ship A \
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\
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\ Observer
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\
Ship B
Label the observer position on the cliff, Ship A, Ship B, and the given angles of depression (44° and 32°).
Step 2: Use trigonometry.
We can use the tangent function (tan) to relate the angles of depression to the distance between the observer and the ships. The tangent of an angle is the opposite side divided by the adjacent side in a right triangle.
In this case, the height of the observer on the cliff (the opposite side) is given as 1000 dm. The adjacent side represents the horizontal distance between the observer and each ship.
For Ship A: tan(44°) = height of observer / distance to Ship A
For Ship B: tan(32°) = height of observer / distance to Ship B
Step 3: Solve the equations.
Let's calculate the distances to Ship A and Ship B separately:
For Ship A: tan(44°) = 1000 / distance to Ship A
Rearranging the equation, distance to Ship A = 1000 / tan(44°)
For Ship B: tan(32°) = 1000 / distance to Ship B
Rearranging the equation, distance to Ship B = 1000 / tan(32°)
Step 4: Calculate the distances.
Using a calculator, evaluate the tangent of 44° and 32°:
For Ship A: tan(44°) ≈ 0.978
distance to Ship A ≈ 1000 / 0.978 ≈ 1022.48 dm
For Ship B: tan(32°) ≈ 0.624
distance to Ship B ≈ 1000 / 0.624 ≈ 1602.56 dm
Step 5: Find the distance between the ships.
The distance between the ships is the difference in their distances from the observer:
distance between ships = distance to Ship B - distance to Ship A
≈ 1602.56 dm - 1022.48 dm
≈ 580.08 dm
Therefore, the distance between the two ships is approximately 580.08 dm.