Imagine a right triangle made with a base of 4 (the distance between the boy and the mom), a height of #x# (the height of the balloon above the mom's eye level), and a hypotenuse of #h# (the straightline distance from the mom's eyes to the balloon). The angle of elevation is defined as the angle located at the mom, measured rising from the eye level.
In order to determine the rate of change, you will need a formula that involves the angle #alpha# being discussed. The simplest one for this problem involves the tangent of #alpha#: #tan(alpha) = x/4#
Differentiate this equation with respect to time #t#:
#sec^2(alpha) (dalpha)/(dt) = 1/4 (dx/dt)#
Given that we were told the balloon is rising at the rate of 1 foot per second, we know that #dx/dt = 1 (ft)/("s")#. This provides us with:
#sec^2(alpha) (dalpha)/(dt) = 1/4 (ft)/s#
#(dalpha)/(dt) = (1/4 (ft)/s)*cos^2(alpha)#
The key now is to determine #cos^2(alpha)#, which becomes relatively easy once we recognize that a balloon rising 1 ft/sec will have risen a total of 3 ft in 3 secs. Thus, at that exact instant, we are describing a right triangle of base 4 and height 3, or a 3-4-5 right triangle. Using trigonometry, we can see that #cos(alpha) = 4/5#
Thus, #(dalpha)/(dt) = (1/4 (ft)/s) * ((4 ft)/(5 ft))^2 = (1/4 (ft)/s) * ((16 cancel("ft"^2))/(25 cancel("ft"^2))) = 4/25 "ft"/s#
