The height of a rocket, launched from the ground, is given by the function y = 30t - 5t^2, where y is in meters and t is in seconds. An observer is sitting at ground level a distance d from the launch point. If r is the distance between the observer and where the rocket is, show that dr/dt = ((30t - 5t^2)(30 - 10t)) / sqrt((30t - 5t^2)^2 + d^2). Show all your work.

Question

The height of a rocket, launched from the ground, is given by the function y = 30t - 5t^2, where y is in meters and t is in seconds. An observer is sitting at ground level a distance d from the launch point. If r is the distance between the observer and where the rocket is, show that dr/dt = ((30t - 5t^2)(30 - 10t)) / sqrt((30t - 5t^2)^2 + d^2). Show all your work.

Answer 1

To find dr/dt, we need to find an expression for r in terms of t.

Let's place the observer at the origin (0, 0) and the launch point at (d, 0).

The distance between the observer and where the rocket is can be found using the distance formula:

r = sqrt((x - d)^2 + y^2)

Since the height of the rocket is given by y = 30t - 5t^2, we can substitute this into the formula:

r = sqrt((x - d)^2 + (30t - 5t^2)^2)

Now, let's find dx/dt and dy/dt.

The x-coordinate of the rocket is changing with time, so dx/dt represents the rate at which the x-coordinate is changing:

dx/dt = d/dt (d) = 0 (since d is a constant)

The y-coordinate of the rocket is given by y = 30t - 5t^2, so dy/dt represents the rate at which the y-coordinate is changing:

dy/dt = d/dt (30t - 5t^2) = 30 - 10t

Now, let's find dr/dt using the chain rule:

dr/dt = d/dt (sqrt((x - d)^2 + (30t - 5t^2)^2))
= (1/2) * (1/sqrt((x - d)^2 + (30t - 5t^2)^2)) * d/dt((x - d)^2 + (30t - 5t^2)^2)

Applying the chain rule:

dr/dt = (1/2) * (1/sqrt((x - d)^2 + (30t - 5t^2)^2)) * (2(x - d) * dx/dt + 2(30 - 10t) * (dy/dt))

Since dx/dt = 0, the first term in the expression becomes 0:

dr/dt = (1/2) * (1/sqrt((x - d)^2 + (30t - 5t^2)^2)) * (2 * 0 + 2(30 - 10t) * (30 - 10t))

dr/dt = (1/2) * (1/sqrt((x - d)^2 + (30t - 5t^2)^2)) * (2 * (30 - 10t)^2)

dr/dt = (1/2) * (1/sqrt((x - d)^2 + (30t - 5t^2)^2)) * (2 * ((30 - 10t)(30 - 10t)))

dr/dt = ((30 - 10t)^2) / sqrt((x - d)^2 + (30t - 5t^2)^2)

Since we are interested in the case where the observer is at the origin (0, 0), we can substitute x = 0 into the expression:

dr/dt = ((30 - 10t)^2) / sqrt((-d)^2 + (30t - 5t^2)^2)

Simplifying further:

dr/dt = ((30 - 10t)^2) / sqrt(d^2 + (30t - 5t^2)^2)

Finally, substituting y = 30t - 5t^2 back into the expression:

dr/dt = ((30 - 10t)^2) / sqrt((30t - 5t^2)^2 + d^2)

Therefore, dr/dt = ((30t - 5t^2)(30 - 10t)) / sqrt((30t - 5t^2)^2 + d^2).