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 4

π
​
, and the angle is changing at a rate of 10
1
​
rad/min. How fast is the balloon rising at this moment?

Answer 1

To determine the rate at which the balloon is rising at the given moment, we can use trigonometry and differentiate the equation. Let's break down the information provided:

Distance between the observer and the lift-off point: 2 miles

Angle between the observer's line-of-sight and the horizontal: 4π radians

Rate of change of the angle: 10/1 rad/min

Let's denote the height of the balloon as h and the time as t.

Using trigonometry, we can establish the relationship between the height of the balloon and the angle:

tan(4π) = h / 2

Differentiating both sides of the equation with respect to time (t), we get:

sec^2(4π) * d(4π)/dt = dh/dt

Since sec^2(4π) is equal to 1 (since the tangent of 4π is undefined), we can simplify the equation to:

d(4π)/dt = dh/dt

Given that the rate of change of the angle is 10/1 rad/min, we can substitute it into the equation:

10/1 = dh/dt

Therefore, the rate at which the balloon is rising at the given moment is 10 miles per minute.