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.
(Use √3 = 1.732)
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.
Let's solve this step-by-step:
Step 1: Draw a diagram to visualize the problem.
Here's how the diagram will look:
```
|
|\
| \ 30°
| \
| \
_______|_____\_____
Car A Balloon Car B
```
Step 2: Identify the given information.
We know that the angles of depression from the balloon to each car are 30° and 45°. The distance between the cars is given as 80 m.
Step 3: Set up an equation.
We can use trigonometry to set up an equation using the tangent function.
Let's consider the angle of depression of 30° first.
tan(30°) = height of balloon / distance from the balloon to Car A
Similarly, for the angle of depression of 45°:
tan(45°) = height of balloon / distance from the balloon to Car B
Step 4: Simplify the equations.
Using the trigonometric identity tan(angle) = opposite/adjacent, we can rewrite the equations.
For the angle of depression of 30°:
1/√3 = height of balloon / distance from the balloon to Car A
For the angle of depression of 45°:
1 = height of balloon / distance from the balloon to Car B
Step 5: Rearrange the equations and solve for the height of the balloon.
We know that the distance between Car A and Car B is 80 m. So, the distance from the balloon to Car A would be 40 m, and the distance from the balloon to Car B would also be 40 m.
For the angle of depression of 30°:
1/√3 = height of balloon / 40
Rearranging the equation:
height of balloon = (1/√3) * 40
height of balloon = (40/√3)
Using the approximation √3 = 1.732, we can calculate the height:
height of balloon = (40 / 1.732) ≈ 23.09 m
Therefore, the height of the balloon is approximately 23.09 meters.