A hot-air balloon is floating above a straight road. To estimate their height above the ground, the balloonists simultaneously measure the angle of depression to two consecutive mileposts on the road on the same side of the balloon. The angles of depression are found to be 16° and 20°. How high is the balloon?
If you draw a diagram, you will see that the height h (in miles) can be found by
h cot16° - h cot20° = 1
h = 1/(cot16° - cot20°)
...
To find the height of the balloon, we can use trigonometry. Let's consider the following diagram:
```
B
/|
/ |h
/ |
/ θ\
/____\
A C
```
Here, A represents one milepost, B represents the balloon, and C represents the other milepost. The angle of depression from the balloon to A is 16°, and the angle of depression from the balloon to C is 20°. We want to find the height of the balloon, which is represented by h.
We know that the angles θ are complementary angles since they both add up to 90°. Therefore, θ = 90° - 20° = 70°.
Now, we can use the tangent function to find the height of the balloon. The tangent of an angle is defined as the ratio between the opposite side and the adjacent side of a right triangle. In our case, the opposite side is h (height of balloon) and the adjacent side is the horizontal distance between the mileposts, let's call it d.
The tangent of an angle is given by the formula:
tan(θ) = opposite/adjacent
In our case:
tan(70°) = h/d
We can rearrange the equation to solve for h:
h = d * tan(70°)
Now, we need to determine the value of d. Since the two mileposts are consecutive, the horizontal distance between them would be the same as the distance between two mileposts, which is 1 mile.
Finally, we can substitute the values in the formula to find the height of the balloon:
h = 1 mile * tan(70°)
To get the numerical value, we can use a scientific calculator or any calculator that has the tangent function. Simply enter 70° and then press the tangent button, followed by multiplying it by 1 mile.
The final result will give us the height of the balloon.