As a hot-air balloon rises vertically, its angle of elevation from a point P on level ground d = 140 kilometers from the point Q directly underneath the balloon changes from 15°10' to 29°30' (see the figure). Approximately how far does the balloon rise during this period? (Give the answer to one decimal place.)
For height at lower angle:
tan 15°10' = height1/140
height1 = 140tan15°10'
in the same way:
height2 = 140tan29°30'
rise of the balloon in that change of angle
= 140tan29°30' - 140tan15°10'
= 41.256
I am sure your units are not correct.
How can they even see the balloon from 140 km away ?
To find the distance the balloon rises, we first need to find the vertical distance between points P and Q.
Using trigonometry, we know that the tangent of an angle is equal to the ratio of the opposite side (vertical distance) to the adjacent side (horizontal distance).
Let's break down the given information:
- The initial angle of elevation is 15°10'.
- The final angle of elevation is 29°30'.
- The horizontal distance between P and Q is 140 kilometers.
To find the vertical distance, we can subtract the initial tangent from the final tangent and multiply by the horizontal distance.
Step 1: Convert the angles to decimal degrees.
Initial angle of elevation: 15°10' = 15 + (10/60) = 15.167°
Final angle of elevation: 29°30' = 29 + (30/60) = 29.5°
Step 2: Calculate the tangent of the initial and final angles.
Initial tangent: tan(15.167°)
Final tangent: tan(29.5°)
Step 3: Subtract the initial tangent from the final tangent and multiply by the horizontal distance.
Vertical distance (rise) = (tan(29.5°) - tan(15.167°)) * 140 km
Let's calculate the vertical distance (rise) using these steps:
Vertical distance = (tan(29.5°) - tan(15.167°)) * 140 km
Vertical distance ≈ 29.5 km (rounded to one decimal place)
Therefore, the balloon rises approximately 29.5 kilometers during this period.
To find the distance the balloon rises during this period, we need to use trigonometry. Let's break down the given information:
- Point P is 140 kilometers away from point Q directly under the balloon.
- The initial angle of elevation from point P is 15°10'.
- The final angle of elevation from point P is 29°30'.
The difference between the initial and final angles of elevation will give us the change in height. We can use the tangent function to find this height. The formula for tangent is:
tangent(angle) = opposite/adjacent
In this case, the opposite side is the change in height, and the adjacent side is the distance from Q to P. Let's calculate the change in height:
Tangent of initial angle = change in height / 140 km
Tangent of 15°10' = change in height / 140 km
To find the change in height, we multiply both sides by 140 km:
change in height = 140 km * tangent of 15°10'
Next, let's calculate the tangent of 15°10':
Tangent of 15°10' = tangent of (15 + 10/60) degrees
Using a scientific calculator or trigonometric tables, we find that the tangent of 15°10' is approximately 0.2679.
Now, we can substitute this value back into the equation:
change in height = 140 km * 0.2679
Calculating this, we find that the change in height is approximately 37.47 kilometers.
Therefore, the balloon rises approximately 37.5 kilometers during this period.