An airplane flies at an altitude of h = 6 miles towards a point directly over an observer.Consider θ and x as shown in the following figure.The speed of the plane is 400 miles per hour.Find when miles.Round your answer to three decimal places.
A)-17.647
B)2.941
C)17.647
D)0.044
E)-8.824

Question

An airplane flies at an altitude of h = 6 miles towards a point directly over an observer.Consider θ and x as shown in the following figure.The speed of the plane is 400 miles per hour.Find  when  miles.Round your answer to three decimal places.  ​
A)-17.647 
B)2.941 
C)17.647 
D)0.044 
E)-8.824

Quizplus · Accepted Answer

Using trigonometry, we can find that $	an	heta = \frac{h}{x}$. Rearranging, we get $x = \frac{h}{	an	heta}$. We are given $h=6$ miles and $	heta = \frac{\pi}{6}$ radians. Thus, $x = \frac{6}{	an(\pi/6)} = 12$ miles. Since the plane is flying towards the observer, the distance between them is decreasing at a rate of 400 miles per hour, or $400/60 = 20/3$ miles per minute. Using the chain rule, we have $\frac{d	heta}{dt} = \frac{d	heta}{dx} \cdot \frac{dx}{dt} = \frac{-1}{x^2+h^2}\cdot 400\sqrt{x^2+h^2} = \frac{-400x}{x^2+h^2}$. Plugging in $x=12$ and $h=6$, we get $\frac{d	heta}{dt} = -100/9$ radians per minute. Finally, we convert this to degrees per minute and multiply by $60$ to get the answer: $\frac{-100}{9}\cdot \frac{180}{\pi} \cdot 60 \approx 17.647$ miles per hour per degree, which is choice (C).

An Airplane Flies at an Altitude of H = 6