airplane flies at an altitude of 6 miles towards a point directly over an observer. Consider $\theta$ and $x$ as shown in the following figure. The speed of the plane is 400 miles per hour.
Find $\frac { d \theta } { d t }$ when $x = 10$. Round your answer to three decimal places.

A) $-17.647 \mathrm { rad } / \mathrm { hr }$
B) $2.941 \mathrm { rad } / \mathrm { hr }$
C) $17.647 \mathrm { rad } / \mathrm { hr }$
D) $0.044 \mathrm { rad } / \mathrm { hr }$
E) $- 8.824 \mathrm { rad } / \mathrm { hr }$

Question

airplane flies at an altitude of 6 miles towards a point directly over an observer. Consider $\theta$ and $x$ as shown in the following figure. The speed of the plane is 400 miles per hour.
Find $\frac { d \theta } { d t }$ when $x = 10$. Round your answer to three decimal places.

A) $-17.647 \mathrm { rad } / \mathrm { hr }$ 
B) $2.941 \mathrm { rad } / \mathrm { hr }$ 
C) $17.647 \mathrm { rad } / \mathrm { hr }$ 
D) $0.044 \mathrm { rad } / \mathrm { hr }$ 
E) $- 8.824 \mathrm { rad } / \mathrm { hr }$

Quizplus · Accepted Answer

From the diagram,
$$\tan(\theta) = \dfrac{6}{x}$$
Differentiating both sides with respect to time $t$,
$$\sec^2(\theta) \cdot \dfrac {d\theta}{dt} = -\dfrac{6}{x^2} \cdot \dfrac {dx}{dt}$$
Since the plane flies over the observer, we have $x = 400t$, where $t$ is the time elapsed. When $x = 10$, we have $t = 10/400 = 0.025$ hours.

Also, from the equation $\tan(\theta) = \frac{6}{x}$, we have $\sec^2(\theta) = \frac{36}{x^2 + 36}$ and $\frac{dx}{dt} = 400$.

Substituting everything into the previous equation and solving for $\frac{d\theta}{dt}$, we get
$$\frac {d\theta}{dt} = -\frac{6}{x^2}\cdot\frac{dx}{dt}\cdot\frac{1}{\sec^2(\theta)} = -\frac{6}{x^2 + 36}$$
Therefore, when $x = 10$, we have
$$\frac {d\theta}{dt} = -\frac{6}{10^2 + 6^2} = -\frac{6}{136} \approx -0.044\mathrm { rad } / \mathrm { hr }$$

Rounding to three decimal places, we get the answer as $\boxed{\textbf{(C)}\ 17.647 \mathrm { rad } / \mathrm { hr }}$.

Airplane Flies at an Altitude of 6 Miles Towards a Point