From the top of the mountain 30m. Above the sea level, the angle of depression of two ships lying in a vertical plane with a mountain top measure 28° and 23°. Find the distance between the ships.
Draw a diagram.
Review your basic trig functions
Now it is clear that the distance x between the ships is
x = 30cot23° - 30cot28°
I made a sketch, showing two right-angled triangles.
One is : base angle 28°, height = 30
tan28 = 30/base
base = 30/tan28
other triangle: let the distance between ships be x
tan23° = 30/(base + x)
base + x = 30/tan23
x = 30/tan23 - base
= 30/tan23 - 30/tan28
= 14.25 m
To find the distance between the ships, we can use the concept of trigonometry and the angles of depression.
Let's assume that the distance between the mountain top and the first ship (Ship A) is x, and the distance between the mountain top and the second ship (Ship B) is y.
From the given information, we know that the angle of depression of Ship A is 28° and the angle of depression of Ship B is 23°. The angle of depression is the angle formed between the line of sight from the observer (mountain top) to the object (ship) and the horizontal line.
For triangle AOB (where O represents the mountain top), we can use trigonometric ratios.
In triangle AOB, the opposite side of angle 28° is x and the adjacent side is 30m+30m=60m (since the distance from the mountain top to the sea level is 30m, and we assume that the ships are also at the sea level).
Using the tangent function, we have:
tan(28°) = x/60
Similarly, in triangle AOC (where O represents the mountain top), the opposite side of angle 23° is y and the adjacent side is 60m (same reason as above).
Using the tangent function, we have:
tan(23°) = y/60
Now we can solve the two equations to find the values of x and y.
Rearranging the first equation, we have:
x = 60 * tan(28°)
Using a calculator, we find that x is approximately 32.948m.
Similarly, rearranging the second equation, we have:
y = 60 * tan(23°)
Using a calculator, we find that y is approximately 26.582m.
Finally, to find the distance between Ship A and Ship B, we can use the formula for finding the distance between two points with coordinates:
distance = sqrt((x1 - x2)^2 + (y1 - y2)^2)
Plugging in the values, we have:
distance = sqrt((32.948 - 26.582)^2) + (0)^2)
Using a calculator, we find that the distance between the ships is approximately 6.366m.
Therefore, the distance between the ships is approximately 6.366m.