a person observed from the top of a light house which is 80 m above the sea level that angle of depression of two ships are 45 degree and 60 degree.

The two ships are on same side of the light house.They are in the line with the foot of the light house.find the distance between the two ships

If the near ship is at distance x and the far ship at distance y,

x/80 = cot60°
y/80 = cot45°

The ships are y-x apart.

y-x = 80(1-1/√3) = 33.8m

not very far apart, eh? And both are about ready to crash into the shore!

These ridiculous scenarios bug me. If we're working with such small distances, don't involve ships at sea! Otherwise, say the angles are 12° and 9° or something that will put the ships far away from shore!

thank u so much

To find the distance between the two ships, we can use trigonometric ratios and the concept of angle of depression.

Let's denote the distance between the foot of the lighthouse and the first ship as 'x', and the distance between the foot of the lighthouse and the second ship as 'y'.

From the given information, we have two angles of depression: 45 degrees and 60 degrees. The angle of depression is the angle formed between the horizontal line and the line of sight from the observer (top of the lighthouse) to the object (ship).

We can draw a diagram to visualize the situation:

A (First ship)
/
80m /
/ x
L /________/_____
Top of lighthouse (O) B (Second ship)
y

In this right-angled triangle, we have the following information:
- OA = 80 m (height of the lighthouse)
- Angle OAB = 45 degrees
- Angle OAC = 60 degrees

Now, let's determine the values of x and y using trigonometric ratios.

For the first ship (A):
In triangle AOB, we have:
tan(OAB) = OA / AB
tan(45) = 80 / x
1 = 80 / x
x = 80

For the second ship (B):
In triangle AOC, we have:
tan(OAC) = OA / AC
tan(60) = 80 / y
√3 = 80 / y
y = 80 / √3

Therefore, the distance between the two ships (distance AB) is given by:
AB = x + y
AB = 80 + 80 / √3

To evaluate this, we can rationalize the denominator by multiplying both the numerator and denominator of the second term by √3:
AB = 80 + (80 * √3 / 3)
AB = (240 + 80√3) / 3

Hence, the distance between the two ships is approximately (240 + 80√3) / 3 meters.