A balloon rises at a rate of 3 meters per second from a point on the ground 30 meters from an observer. Find the rate of change of the angle of elevation (in radians per second or rad/sec) of the balloon from the observer when the balloon is 30 meters above the ground.
1/20
To find the rate of change of the angle of elevation of the balloon from the observer, we need to use trigonometry and differentiation.
Let's denote the angle of elevation as θ at a certain point in time when the balloon is h meters above the ground, and let t represent time in seconds.
We are given that the balloon is rising at a rate of 3 meters per second, which means the rate of change of the height of the balloon with respect to time (dh/dt) is 3 m/s.
Now, let's create a right-angled triangle to represent the situation. The vertical side represents the height of the balloon (h), the horizontal side represents the distance between the observer and the point on the ground directly below the balloon (30 meters), and the hypotenuse represents the line of sight between the observer and the balloon.
Using trigonometry, we can find the relation between θ, h, and 30 meters:
tan(θ) = h / 30
To find the rate of change of θ with respect to time (dθ/dt), we need to differentiate both sides of this equation with respect to time, using the chain rule.
Differentiating tan(θ) = h / 30:
sec^2(θ) * dθ/dt = dh/dt / 30
Substituting dh/dt = 3 m/s and rearranging the equation, we get:
dθ/dt = (3 / 30) * sec^2(θ)
Now, we need to find the value of θ when the balloon is 30 meters above the ground. At this point, h = 30, and we can solve for θ using the equation tan(θ) = h / 30:
tan(θ) = 30 / 30
tan(θ) = 1
θ = arctan(1)
Finally, substitute θ = arctan(1) into the equation dθ/dt = (3 / 30) * sec^2(θ) to calculate the rate of change of the angle of elevation:
dθ/dt = (3 / 30) * sec^2(arctan(1))
Using the identity sec^2(arctan(x)) = 1 + x^2 and substituting x = 1, we get:
dθ/dt = (3 / 30) * (1 + 1^2)
dθ/dt = (3 / 30) * 2
dθ/dt = 1/5 rad/sec
Therefore, the rate of change of the angle of elevation of the balloon from the observer when the balloon is 30 meters above the ground is 1/5 rad/sec.