Question

A weather balloon is rising vertically at the rate of 5 ft/s. An observer is standing on the ground 400 ft horizontally from the point where the balloon was released. What is the vertical speed of the observer's line of sight?

a) 0 ft/s
b) 5 ft/s
c) 10 ft/s
d) 20 ft/s

Answer 1

The vertical speed of the observer's line of sight toward a rising weather balloon is not directly provided by any of the options. By reasoning through the provided choices and understanding that the rate cannot be zero, nor can it exceed the rising rate of the balloon, and that it must be changing as the balloon ascends, option (b) 5 ft/s is determined to be the most reasonable provided answer.

Step-by-step explanation:

To determine the vertical speed of the observer's line of sight as a weather balloon ascends vertically at a rate of 5 ft/s, we use a bit of right triangle trigonometry and the Pythagorean theorem. At any moment, the observer, the balloon, and the point on the ground vertically beneath the balloon form a right triangle.

We can consider the horizontal distance from the observer to the point beneath the balloon as one leg (400 ft), the vertical distance the balloon has traveled as the other leg (which increases at 5 ft/s), and the line of sight as the hypotenuse.

Let's denote the vertical distance the balloon has ascended as 'v(t)', the horizontal distance as a constant '400' ft, and the line of sight as 's(t)'. As the balloon rises, the line of sight length changes and the rate at which it changes is the vertical speed of the observer's line of sight, which can be found by differentiating 's(t)' with respect to time.