Calculus Triangle Related Rate

Question

So I have a problem where there is a right triangle with the height (4500ft) and the rate at which the angle is changing (pi/180radians per second) and I need to find the ground speed.

I figured out that the ground speed dx/dt=-4500csc^2(theda) d(theda)/dt

But after this I am stuck. I really need help because I just don't get how I'm supposed to find the speed, please help as soon as possible I am forever thankful.

Also I kind of got an answer of 1272 mph, is there any chance that is even possibly right?

Answer 1

Sara,

Due to lack of complete information, I need to make several guesses in order to answer this question.  

Let's suppose the right triangle contains points A, B, C with BC being the horizontal base and C is to the right of B and AC is the hypotenuse of the right triangle.  ∠BCA=θ.

 

You said θ is changing at π/180 radians/sec.  This tells me that C is moving towards B.  Since the distance between B and C is changing, let's call this variable x

 

With the above information, we can write the following equation: dθ/dt = π/180.  We also know that the height of the triangle is 4500'.  Using that information we can write the following equation.

 

tan(θ) = 4500/BC = 4500/x.  Or x = 4500cot(θ)

 

dx/dt = -4500csc²(θ).dθ/dt.  The negative value for dx/dt indicates that x is decreasing.  From this equation it is clear that dx/dt changes with θ and dθ/dt.  But since dθ/dt is constant, dx/dt only depends on θ.  The speed is variable.  The last missing piece of information that is essential for calculating the ground speed is the value of θ.  If we assume that the value of θ is π/2 (when C coincides with B and moving to the left) the ground speed is -4500csc²(θ).dθ/dt=-25π feet/sec

Answer 2

From what we have,

 

dx/dt = -4500ft csc²θ dθ/dt = -4500ft csc²(π/2) (π/180 sec) = -78.54 ft/sec

 

 

 

This is your velocity.  To get the speed in mph, do unit conversions and take the absolute value:

 

v = (78.54 ft/sec)(1 mile/5280 ft)(3600 sec/1 hour) = 53.5 mph