A rocket is rising vertically at a rate of 5400 mph. An observer on the ground is standing 20
miles from the rocket’s launch point. How fast in radians per second is the angle of elevation
between the ground and the observer’s line of sight of the rocket increasing when the rocket
is at an elevation of 40 miles.
Solution:
h
tan θ = Rocket
20
d tan θ dθ
= Sec 2θ
dt dt
dθ 1 dh
Sec 2θ =
dt 20 dt h
dh
= 5400 mph
dt
Observer
dh 5400
=
dt 3600 20
dh 3
= miles/sec.
dt 2
dθ 1 dh
Sec 2θ =
dt 20 dt
when h = 40
40 2√5
tan θ = 4
20
2 5 θ
Sec θ =
2 2
Sec 2θ = 5
dθ 1 ⎛ 3 ⎞
5 = ⎜ ⎟
dt 20 ⎝ 2 ⎠
dθ 3
= rad / sec.
dt 200
, Refresher - MATHEMATICS Quiz 20
PROBLEM 2:
A grain-elevator hopper is to be constructed with a circular cylinder of
radius 2 ft. and altitude of h ft. on top of a right circular cone whose
altitude is one half that of the cylinder. What value of h will make the total
volume of the hopper equal to 500 ft3?
Solution: r =2 r =2
π(2) 2 (h / 2)
V = π(2) h + 2
3
2
500 = 4π h + π h h
3
h = 34.1 ft.
h/2
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