A hot-air balloon is rising vertically at a constant speed, an observer at a distant observes the
elevation angle to be 30° at 10:00am, at 10:10am the elevation angle becomes 34°, then at 10:30am the
elevation angle of the balloon should be closest to (using the table below) (A) 34° (B) 39° (C) 41° (D)
42° (E) 43°
θ 34° 39° 40° 41° 42° 43°
sinθ /0.559 /0.629 /0.643 /0.656 /0.669 /0.682
cosθ /0.829 /0.777 /0.766 /0.755 /0.743 /0.731
tanθ /0.675 /0.810 /0.839 /0.869 /0.900 /0.933
If the balloon's height at 10:00 is h, and it rises z in 10 minutes, then if the distance of the observer is x we have
at 10:00 h/x = tan30°
at 10:10 (h+z)/x = tan34°
eliminating x, we have
h/tan30° = (h+z)/tan34°
z = h(tan34°/tan30° - 1)
so at 10:30 the balloon has risen another 2z and we have the new angle m using
tan m = (h+3z)/x = (h + 3(h(tan34°/tan30° - 1)))/(h/tan30°)
tan m = (1 + 3(√3 tan34° - 1))/√3 = 0.8688
m = 40.98°