Question

A hot air ballon rising vertically is tracked by an observer located 3 miles from the lift-off point. At a certain moment, the angle between the observer's line-of-sight and the horizontal is π 5 , and it is changing at a rate of 0.1 rad/min. How fast is the balloon rising at this moment?

Answer 1

Answer:

0.458 mi/min

Step-by-step explanation:

We are given that

Distance of observer from lift-off point=3 miles

$\theta=\frac{\pi}{5}$

$\frac{d\theta}{dt}=0.1rad/min$

We have to find the rate at which the balloon rising at this moment.

$tan\theta=\frac{perpendicular\;side}{base}$

Using the formula

$tan\theta=\frac{h}{3}$

Differentiate w.r.t t

$sec^2\theta\frac{d\theta}{dt}=\frac{1}{3}\frac{dh}{dt}$

Using the formula

$\frac{d(tan\theta)}{d\theta}=sec^2\theta$

$\frac{dh}{dt}=3sec^2\theta\frac{d\theta}{dt}$

$\frac{dh}{dt}=3sec^2(\frac{\pi}{5})\times 0.1$

$\frac{dh}{dt}=0.458 mi/min$