• Support PF! Buy your school textbooks, materials and every day products Here!

Angle of Elevation Related Rates help

  • Thread starter B3NR4Y
  • Start date
  • #1
B3NR4Y
Gold Member
170
8
I need help with this problem in my calculus book.
An airplane at an altitude of 10,000 feet is flying at a constant speed on a line which will take it directly over an observer on the ground. If, at a given instant, the observer notes that the angle of elevation of the airplane is 60 degrees and is increasing at a rate of one degree a second, find the speed of the airplane. (Hint: tangent of theta is equal to sine of theta over cosine of theta)

I've worked it and had tangent of theta equals "x" over 10,000. And took the derivative of each side to get secant squared of theta equals one over 10,000 times dx/dt. I solve for dx/dt with my information and get 40,000 feet per second. However the boom says 10,000*pi over 135.

I don't get where the pi comes from.

Sorry for no LaTeX, I'm on my phone.
 

Answers and Replies

  • #2
6,054
390
The full derivative of the angle term must also include the derivative of the angle, i.e., the angular speed. Whose value is given in degrees per second, but it must be converted to the radian measure.
 
  • #3
B3NR4Y
Gold Member
170
8
Thank you. The reason I didn't include it was because it was 1 degree a second, but what you're saying is I should convert all angles to radian? I'll try that...
 
  • #4
B3NR4Y
Gold Member
170
8
I still can't seem to get it. Can someone tell me how the book got that answer? I don't know why it would give me the hint it did if I don't really need it.
 
  • #5
6,054
390
What is the formula you got?
 
  • #6
B3NR4Y
Gold Member
170
8
If you want to know what hint they gave: The tangent of theta being equal to the sine of theta over the cosine of theta.

If you want to know the answer I got: (40,000*pi)/180
I know it isn't right , so I didn't feel the need to simplify. The formula I get for the rate x changes with respect to time is:
dx/dt = pi/180 * sec^2 (pi/3)*10,000
 
  • #7
B3NR4Y
Gold Member
170
8
If you want to know what hint they gave: The tangent of theta being equal to the sine of theta over the cosine of theta.

If you want to know the answer I got: (40,000*pi)/180
I know it isn't right , so I didn't feel the need to simplify. The formula I get for the rate x changes with respect to time is:
dx/dt = pi/180 * sec^2 (pi/3)*10,000
\begin{equation}
\frac{dx}{dt} = \frac{\pi sec^{2}(\frac{\pi}{3}) 10,000}{180}
\end{equation}

\begin{equation}
tan \theta =\frac{x}{10000}
\end{equation}
\begin{equation}
\frac{d}{dt} (tan \theta) = \frac{d}{dt} (\frac{x}{10000})
\end{equation}
\begin{equation}
sec^{2} \theta \frac{d\theta}{dt} = \frac{1}{10000} \frac{dx}{dt}
\end{equation}
\begin{equation}
10,000 (sec^{2} (\frac{\pi}{3}) \frac{\pi}{180}) = \frac{dx}{dt}
\end{equation}
\begin{equation}
10,000 (\frac{4\pi}{180})
\end{equation}
\begin{equation}
\frac{40,000\pi}{180}
\end{equation}
\begin{equation}
\frac{2,000\pi}{9}
\end{equation}

Where have I steered wrong
 
Last edited:
  • #8
B3NR4Y
Gold Member
170
8
Oh, and the hint they gave was

\begin{equation}
tan\theta = \frac{sin\theta}{cos\theta}
\end{equation}
 
  • #9
6,054
390
## \tan \theta = \frac x {10000} ## is wrong. The correct formula is ## \tan \theta = \frac {10000} x ##.
 
  • #10
B3NR4Y
Gold Member
170
8
So
\begin{equation}
sec^{2} \theta \frac{d\theta}{dt} = -\frac{10,000}{x^{2}} \frac{dx}{dt}
\end{equation}
 
  • #11
6,054
390
Yes. Eliminate x from the expression and then find v. Alternatively, you could use cot x = x/10000.
 
  • #12
B3NR4Y
Gold Member
170
8
Thanks :D I got the correct answer. You are a calculus god, have the ceremonial loin cloth.
 

Related Threads for: Angle of Elevation Related Rates help

Replies
6
Views
18K
  • Last Post
Replies
1
Views
2K
Replies
1
Views
8K
  • Last Post
Replies
2
Views
7K
  • Last Post
Replies
7
Views
2K
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
1
Views
1K
Top