Rate of change if the angle...?

Question

A helicopter rises at the rate of 8 feet per second from a point on the ground 60 feet from an observer.

Find the rate of change of the angle of elevation when the helicopter is 25 feet above the ground.

Explain your work, and good luck! :)

Answer 1

Let angle of elevation be x and height be h ft
tan x = h / 60
x = tan^(-1) (h / 60)
Let h / 60 = y
x = tan^(-1).y
dx/dy = 1 / (1 + y²)
dy/dh = 1 / 60
dx/dh = (dx/dy).(dy/dh)
dx/dh = (1 / (1 + y²)).(1/60)
dx/dh = (1 / (1 + h²/60²)).(1/60)
dx/dh = 1 / (60 + h²/60)
dx/dh = 60 / (3600 + h²)
dx/dh = 60 / (3600 + 25²)
dx/dh = 0.0124 radians/sec

Answer 2

When the helicopter is 25 ft up, the angle is tan^-1(25/60) = .39479 radian. One second later the angle will become tan^-1(33/60) = .50284 rad, so the rate is (.50284 - .39479) = .10805 r/s.

Answer 3

ok, it relatively is going to help you in case you first draw a precise triangle with a base of 60 ft. the splendid area of the triangle ought to arise x ft., and the perspective interior the decrease left hand nook must be theta. ok, so which you be responsive to that helicopter is increasing at 8 ft. in line with 2nd (dx/dt in different words) and you be responsive to that theta relies upon on the place the helicopter is. now you are able to relate the two by right here equation: theta = tan^-a million(x/60) (this ought to make experience in case you do the trig) Now we ought to take the by-product of theta to work out how theta adjustments with time. because of the fact the by-product of tan^-a million is a million/(a million+x^2), the by-product of theta turns into right here utilising the chain rule: dtheta/dt = (dx/dt) / ( 60 (a million + (x/60)^2) ) Simplifying a splash supplies us: dtheta/dt = 3600(8) / (216000 + 60x^2) (I plugged in dx/dt = 8 right here and prolonged the num and denom by 60^2) we've got a function that relates the exchange in perspective to the top of the helicopter. So all we ought to do is substitute 25 ft. in for x and we are carried out. i'm getting 0.1136 rad/sec or approximately 6.5 levels/sec. i think of it relatively is right, even though it relatively is been awhile because I even have carried out those products.