From an aeroplane, at an altitude of 2.3km, the angle of depression of the airport is 32Degrees and the angle of depression of the town is 18degrees (both are in direct line ahead of the plane) How far is the town from the airport?
If we label the diagram as follows:
P = plane
A = point on ground just below the plane
B = airport
C = town
we want to find BC
2.3/AB = tan 32
2.3/(AB+BC) = tan 18
eliminating the AB we have
2.3/tan32 = 2.3/tan18 - BC
Now just solve for BC.
To find the distance between the town and the airport, we can use trigonometric ratios, specifically tangent.
Let's assume that the distance between the airplane and the airport is x kilometers, and the distance between the airplane and the town is y kilometers.
Given that the angle of depression of the airport is 32 degrees, we can draw a right triangle with the airplane as the top vertex, the base of the triangle representing the distance between the airplane and the airport.
Using the tangent ratio, we can set up the following equation:
tan(32) = 2.3 / x
To find the distance between the airplane and the town, we need to find the length of the base of the triangle formed between the airplane, town, and the point on the ground directly below the airplane.
We can use the information about the angle of depression of the town, which is 18 degrees. Again, we can set up a right triangle with the airplane as the top vertex and the base representing the distance between the airplane and the town.
Using the tangent ratio once again, we can set up the following equation:
tan(18) = 2.3 / (x + y)
Now we have two equations:
1) tan(32) = 2.3 / x
2) tan(18) = 2.3 / (x + y)
We can solve this system of equations to find the values of x and y.
First, let's solve equation (1) for x:
x = 2.3 / tan(32)
Next, substitute this value of x into equation (2) and solve for y:
tan(18) = 2.3 / (2.3 / tan(32) + y)
Now, we can simplify the equation:
tan(18) = 1 / (1 / tan(32) + y / 2.3)
Next, convert tan(18) and tan(32) into decimal values using a calculator.
Finally, solve for y:
y = (2.3 / tan(32)) * (1 - (1 / tan(18)))
Using this equation, you can find the value of y, which represents the distance between the airplane and the town.