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

Ask by Johnson Vaughan. in the United States

Mar 15,2025

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

Ask by Johnson Vaughan. in the United States

Mar 15,2025

UpStudy AI · Accepted Answer

Let $$ \theta $$ be the angle of elevation, and let $$ h $$ be the height of the balloon. The horizontal distance from the observer to the lift-off point is 2 miles. We have the relation:
$$
\tan \theta = \frac{h}{2}
$$
Thus, the height of the balloon is:
$$
h = 2 \tan \theta
$$

Differentiate both sides with respect to time $$ t $$:
$$
\frac{dh}{dt} = 2 \sec^2 \theta \cdot \frac{d\theta}{dt}
$$

We are given:
$$
\theta = \frac{\pi}{3} \quad \text{and} \quad \frac{d\theta}{dt} = 0.1 \, \text{rad/min}
$$
At $$\theta = \frac{\pi}{3}$$, we have:
$$
\cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \quad \Rightarrow \quad \sec\left(\frac{\pi}{3}\right) = 2 \quad \Rightarrow \quad \sec^2\left(\frac{\pi}{3}\right) = 4
$$

Substitute into the differentiated equation:
$$
\frac{dh}{dt} = 2 \times 4 \times 0.1 = 0.8 \, \text{miles/min}
$$

Thus, the balloon is rising at a rate of $$ 0.8 \, \text{miles/min} $$.
The balloon is rising at a rate of 0.8 miles per minute.

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

Upstudy AI Solution

Answer

Solution

Bonus Knowledge

Related Questions

Latest Calculus Questions

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

Upstudy AI Solution

Answer

Solution

Bonus Knowledge

Related Questions

Latest Calculus Questions

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