Question

A balloon is released 15 feet away from an observer. The balloon is rising vertically at. a rate of 2 ft/sec and at the same time the wind is carrying it horizontally away from the observer at a rate of 3 ft/sec. At what speed is the angle of inclination of the observer’s line of sight changing 5 seconds after the balloon is released?

Answer 1

To determine the rate of change of the observer's line of sight angle, we must use trigonometric relationships and the concept of related rates in calculus, involving differentiating with respect to time.

Step-by-step explanation:

The question involves calculating the rate at which the angle of inclination of an observer's line of sight is changing as a balloon rises and moves horizontally away. This requires an application of related rates, a concept in differential calculus. To determine the rate of change of the angle, we use trigonometric relationships and the Pythagorean theorem.

Let x represent the horizontal distance of the balloon from the observer and y represent the vertical distance from the observer to the balloon. The angle of inclination is \( heta\). After 5 seconds, the balloon has risen 5 \(\times\) 2 ft = 10 ft and has also moved horizontally away 5 \(\times\) 3 ft = 15 ft. Initially, the balloon was 15 feet away horizontally, so the total horizontal distance is x = 15 ft + 15 ft = 30 ft and the vertical distance is y = 10 ft.

Using the relationship \(tan(\theta) = y/x\), we differentiate both sides with respect to time, t, to find the rate of change of the angle, d\(\theta\)/dt. With the given rates of rise and horizontal movement, and knowing x, y, we can solve for d\(\theta\)/dt.

Answer 2

s = (v-u)/2 * t, where v is the final velocity, u is the initial velocity, s is the displacement and t is the time.

40 = (v-2.8)/2 * 8.5
4.71 = v-2.8/2
9.41 = v- 2.8
v = 12.2 m/s

v^2 = u ^2 +2as where v is the final velocity, u is the initial velocity, a is the acceleration and s is the displacement.

12.2^2 = 2.8^2 + 2a(40)
80a = 12.2^2 - 2.8^2
80a = 148.84 - 7.84
80a = 141
a = 1.7625m/s^2