from a top of a vertical tower the angles of depression of two cars in the same straight line with the base of the tower at an instant are found to Ber 45 degree and 60 degree if the cars are 100 m apart and are on the same side of the tower find the height of the tower
236.5
review the definition of cot(x). Draw a diagram and you can see that
h*cot45 - h*cot60 = 100
To find the height of the tower, we can use the tangent function.
Let's visualize the situation.
We have a tower, and from the top of the tower, two cars in the same straight line with the base of the tower are observed. The angles of depression of the two cars are 45 degrees and 60 degrees, respectively. The cars are 100 meters apart and on the same side of the tower.
Let's assign some variables:
Let H be the height of the tower.
Let x be the distance from the base of the tower to the first car.
Let y be the distance from the base of the tower to the second car.
Since the two cars are on the same side of the tower, we know that the distance from the base of the tower to the second car is equal to the distance from the base of the tower to the first car plus the distance between the two cars:
y = x + 100
Now, let's use the tangent function to relate the angles of depression to the distances and the height of the tower:
tan(45 degrees) = H / x
tan(60 degrees) = H / y = H / (x + 100)
We have two equations, so let's solve them simultaneously.
First, let's solve for H in terms of x using the first equation:
H = x * tan(45 degrees)
Now, substitute this value of H into the second equation:
tan(60 degrees) = (x * tan(45 degrees)) / (x + 100)
Now, we can solve for x:
tan(60 degrees) = (x * tan(45 degrees)) / (x + 100)
(x + 100) * tan(60 degrees) = x * tan(45 degrees)
(x + 100) * √3 = x * 1
Simplifying this equation:
√3x + 100√3 = x
100√3 = (1 - √3)x
x = 100√3 / (1 - √3)
Now, let's substitute this value back into the equation for H:
H = x * tan(45 degrees)
H = (100√3 / (1 - √3)) * 1
Simplifying this:
H = 100√3 * (1 + √3)
So the height of the tower is 100√3 * (1 + √3) meters.