A hot-air balloon is rising vertically at a constant speed, an observer at a distant observes the elevation angle to be 30° at 10:00am, at 10:10am the elevation angle becomes 34°, then at 10:30am the elevation angle of the balloon should be closest to (using the table below) (A) 34° (B) 39° (C) 41° (D) 42° (E) 43°
change in angle / change in time = 4 deg/ 10 min
change in time = 10:30 - 10:10 = 20 min
so
change in angle if linear = (4 deg / 10 min) * 20 min = 8 degrees
34 + 8 = 42 degrees
a constant vertical speed does not yield a constant change in θ.
It yields a constant change in tanθ
To solve this problem, we need to use the concept of similar triangles.
Given that the hot-air balloon is rising vertically at a constant speed, we can assume that the observer and the balloon are forming a right triangle with the ground. Let's call the distance between the observer and the balloon "d" and the altitude of the balloon "h".
First, we need to find the trigonometric relationship between the angle of elevation and the sides of the triangle. The tangent of an angle of elevation is equal to the quotient of the length of the side opposite the angle (h in this case) and the length of the side adjacent to the angle (d in this case).
Using this concept, we can set up the following equation to find the initial altitude of the balloon:
tan(30°) = h / d
Next, we can set up another equation to find the altitude of the balloon at 10:10am:
tan(34°) = h / d
Since the balloon is rising vertically at a constant speed, the change in altitude of the balloon between 10:00am and 10:10am will be the same as the change in altitude between 10:10am and 10:30am.
Therefore, we can subtract the initial altitude of the balloon from the altitude at 10:10am to get the change in altitude, and then add that change to the altitude at 10:10am to find the altitude at 10:30am.
Let's calculate it:
At 10:00am:
tan(30°) = h / d
At 10:10am:
tan(34°) = h / d
Let's find the value of h/d for both cases:
For 10:00am:
tan(30°) = h / d
h/d = tan(30°)
For 10:10am:
tan(34°) = h / d
h/d = tan(34°)
Now, let's find the change in altitude between 10:00am and 10:10am:
Change in altitude = (h/d at 10:10am) - (h/d at 10:00am)
Now, add this change in altitude to the altitude at 10:10am to get the altitude at 10:30am:
Altitude at 10:30am = (h/d at 10:10am) + (Change in altitude)
Now we can substitute the values and calculate the altitude at 10:30am:
Altitude at 10:30am = (tan(34°)) + (Change in altitude)
Finally, we compare the value we calculated for the altitude at 10:30am with the values provided in the given answer choices (A) 34°, (B) 39°, (C) 41°, (D) 42°, and (E) 43°.
The answer choice with the closest value to our calculated altitude at 10:30am should be the correct answer.