An observer is 5oo ft from launch site of a rocket. at the moment that the angle of elevation is pi/4 radians, the angle is increased threat of 0.2 rad/min. How fast is the rocket rising that moment?

Answer 1

We are being asked to find dh/dt

We are given that dθ/dt = 0.2 rad/min

We can say that tan(θ) = h/500 or h = 500tan(θ)

We can take the derivative of that equation with respect to time (t) making sure we use the chain rule:

Plugging in θ = π/4 and dθ/dt = 0.2 rad/min we get:

dh/dt = 500sec²(π/4)(0.2)

Since cos(π/4) = √2/2 (aka 1/√2) then sec(π/4) = √2 and sec²(π/4) = 2 so

dh/dt = 500(2)(0.2) = 200 ft/min

Answer 2

According to the problem, we have the drawing above, the y is the height of the rocket and the x is distance form the observing site to the rocket launch site. And we have relationship of:

,

so we have:

.

Now apply the value of x, θ and dθ/dt given by the question you will get he speed of the rocket as dy/dt.

Answer 3

Hi Anna F.,

A few things.

1) You should proofread your questions before submitting them! I don't, for example, see an increased threat from any particular angle! But I'll assume you meant, "the angle is increasing at the rate of".

2) The trajectory of the rocket is unspecified, and that is crucial to a solution. If the rocket were pointed away at a line following an asymptote of pi/4 radians with respect to the observer, it could be rising indefinitely fast and yet not increasing its apparent angle to the observer at all! So, let's assume the simplest case, that the rocket is launched exactly vertically and follows a vertical line of ascent.

3) So now you may generate an equation of the form of angle = sin (y/x) , where y is the vertical height of the rocket, and x is the distance observer<->launch_site (500ft) . Can you differentiate that equation with respect to time on both sides of the equals sign? You will need to express y in terms of its present value (solve the triangle specified) plus a velocity term (first degree with respect to time), and differentiate the sin function (use the chain rule for the expression within the sin function). Then plug in your stated values, and crank out the answer (the coefficient of the velocity term).

-- Cheers, --Mr. d.