Final answer:
The question involves related rates in calculus to find the rate at which the observer's eye must rotate to track a rising balloon. When the balloon is 50 ft above, the observer's eye rotation rate is 1/100 rad/sec.
Step-by-step explanation:
The problem described is an application of related rates, a concept in calculus used to determine how rates change with each other. In this case, we are attempting to find the rate at which the observer's eye is rotating upward (theta rate) to keep the balloon in sight. This rate will be in radians per second.
To solve this, we can use the right triangle formed by the observer, the point of observation, and the balloon. Here, the vertical side is the balloon height (opposite side), the horizontal side is the distance from observer to balloon launch point (adjacent side), and the angle of elevation theta is the angle from the observer's eye line perpendicular to the ground to the line of sight to the balloon.
Using the tangent function will create the following relationship: tan(theta) = opposite/adjacent = balloon height/distance from observer. Differentiating this relationship with respect to time t and using the given rates, we can solve for the rate at which the angle theta is changing. The rate of change of balloon height is 10 ft/sec, and the distance from observer is 100 ft. When the balloon is 50 ft above the observer's eye level, theta rate can be found using related rates.
Therefore, at the moment when the balloon is 50 ft above, the observer's eye rotation speed in radians per second must be calculated based on the given numerical values and the tangent function's differentiation which yields the result: theta rate = 1/100 rad/sec.