Question
A hot air balloon rising vertically is tracked by an observer located 2 miles from the lift-off point. At a certain moment, the angle between the
observer’s line-of-sight and the horizontal is
, and it is changing at a rate of
. How fast is the balloon rising at this moment?
miles
observer’s line-of-sight and the horizontal is
miles
Ask by Johnson Vaughan. in the United States
Mar 15,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The balloon is rising at a rate of 0.8 miles per minute.
Solution
Let
be the angle of elevation, and let
be the height of the balloon. The horizontal distance from the observer to the lift-off point is 2 miles. We have the relation:
Thus, the height of the balloon is:
Differentiate both sides with respect to time
:
We are given:
At
, we have:
Substitute into the differentiated equation:
Thus, the balloon is rising at a rate of
.
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Bonus Knowledge
To find how fast the balloon is rising, we can use related rates in calculus. Let
be the height of the balloon above the ground, and
be the angle between the observer’s line of sight and the horizontal.
From the observer’s perspective, using basic trigonometry in a right triangle, we have:
Differentiating both sides with respect to time
gives us:
We know:
First, let’s calculate
:
Now substitute the values into the differentiated equation:
To find
:
So, the balloon is rising at a rate of
miles per minute.