From a balloon, vertically above a straight road, the angles of depression of two cars at an instant are found to be 30° and 45° If the cars are 80 m apart, find the height of the balloon.

Question

(Use √3 = 1.732)​

Answer 1

Let's denote the height of the balloon as h.

From the position of the balloon, we can form a right triangle with the road as the horizontal leg and the height of the balloon as the vertical leg.

Let's consider the car closer to the balloon. The angle of depression from the balloon to this car is 30°. This means that the angle between the horizontal leg and the line connecting the car to the balloon is also 30°.

Similarly, for the farther car, the angle of depression is 45°. This means that the angle between the horizontal leg and the line connecting the car to the balloon is also 45°.

Now, let's consider the triangle formed by the balloon, the closer car, and the road. In this triangle:

tan(30°) = h/x, where x is the horizontal distance between the balloon and the closer car.

Similarly, for the triangle formed by the balloon, the farther car, and the road:

tan(45°) = h/(x + 80), where x + 80 is the horizontal distance between the balloon and the farther car.

For the first equation, we have:
h/x = tan(30°)

For the second equation, we have:
h/(x + 80) = tan(45°)

Now, let's solve these two equations to find the value of h:

h/x = 1/(√3)
h/(x + 80) = 1

From the second equation, we have:
h = x + 80

Substituting this value of h into the first equation, we get:
(x + 80)/x = 1/(√3)

Cross-multiplying, we have:
(x + 80)(√3) = x

Expanding, we get:
√3x + 80√3 = x

Bringing all the x terms to one side, we have:
x - √3x = -80√3

Factor out x:
x(1 - √3) = -80√3

Dividing both sides by (1 - √3):
x = -80√3 / (1 - √3)

To rationalize the denominator, we multiply both the numerator and the denominator by (1 + √3):

x = (-80√3 / (1 - √3)) * ((1 + √3) / (1 + √3))
x = (-80√3 * (1 + √3)) / (1 - 3)
x = (-80√3 - 240) / (-2)
x = 40√3 + 120

Now, let's plug this value of x into the equation h = x + 80:

h = (40√3 + 120) + 80
h = 40√3 + 200

Therefore, the height of the balloon is 40√3 + 200 meters.