A spy tracks a rocket through a telescope to determine its velocity. The rocket is traveling vertically from a launching pad located 10 km away. At a certain moment, the spy's instruments show that the angle between the telescope and the ground is equal to pi/3 and is changing at a rate of 0.5 radians/min. What is the rocket's velocity at that moment?
tan T = h/10
so
h = 10 tan T
dh/dt = 10 d(tan T)/ dt = 10 (sec^2 T) dT/dt
so
dh/dt = 10 (1/cos^2 pi/3) (.5)
but
cos pi/3 = cos 60 = sin 30 = 1/2
so
dh/dt = 10(4)(.5) = 20 km/min = 20,000 m/min
= 20,000 m/min *1 min/60s = 333 m/s
See, they are all the same.
I'am very happy so much thank's
Well, well, well, look who's playing James Bond with their spy gadgets! Quite a thrilling scenario you've got here.
Let's break it down, shall we? We know that the spy is observing the rocket from a distance of 10 km, and the angle between the telescope and the ground is changing at a rate of 0.5 radians/min. This means that the rocket is moving at an angle of pi/3, and that angle is changing with respect to time.
To find the rocket's velocity, we can start by drawing a right triangle. The base of the triangle would be the horizontal distance between the launching pad and the spy, which is 10 km. The height of the triangle would represent the vertical distance the rocket has traveled at that moment.
Since we're given the rate at which the angle is changing, we can relate it to the rocket's velocity using a trigonometric function. The tangent of the angle is equal to the ratio of the velocity to the horizontal distance. So we have:
tan(pi/3) = velocity / 10 km
Now we can solve for the velocity:
velocity = 10 km * tan(pi/3)
Now, I don't have a calculator handy, so you'll have to punch in those numbers yourself. But don't worry, math is no rocket science... well, except in this case. Good luck, Agent 007!
To determine the rocket's velocity at a certain moment, we first need to understand the situation and make some relevant observations:
1. The angle between the telescope and the ground is given as π/3 radians.
2. The angle is changing at a rate of 0.5 radians/min.
3. The rocket is traveling vertically upward.
To find the rocket's velocity, we can utilize trigonometry and calculus. Here's how we can approach the problem step by step:
Step 1: Draw a diagram representing the situation described. Label the relevant components.
|
| <-- Telescope
|
|\
| \
10| \ <-- Launching Pad
|___\
10 km
Step 2: Let's define some variables:
- Let R be the distance between the rocket and the telescope.
- Let θ be the angle between the telescope and the ground.
- Let t be the time elapsed.
Step 3: Use trigonometry to express R in terms of θ. In our case, from the given information, we have a right triangle where the adjacent side is 10 km and the angle opposite to it is θ. So, we have:
R = 10 km / cos(θ)
Step 4: Differentiate R with respect to time t to get the rate of change of R.
dR/dt = (dR/dθ) * (dθ/dt) (Using the chain rule of differentiation)
Now, we need to find dR/dθ, i.e., the derivative of R with respect to θ.
dR/dθ = d(10 km / cos(θ)) / dθ
To compute this derivative, we can use the quotient rule:
dR/dθ = [(-10 km * sin(θ)) * cos(θ)] / (cos^2(θ))
= -10 km * tan(θ)
Substituting this result back into our equation for dR/dt:
dR/dt = (-10 km * tan(θ)) * (dθ/dt)
Step 5: Substitute the given values into the equation to find the rocket's velocity when θ = π/3 and dθ/dt = 0.5 radians/min.
dR/dt = (-10 km * tan(π/3)) * (0.5 radians/min)
Simplifying further,
dR/dt = -5 km/sqrt(3) radians/min
So, the rocket's velocity at that specific moment is -5 km/sqrt(3) radians/min, with the negative sign indicating that the rocket is moving downwards.