Calculus airplane related rates problem ( cosine rule)

Question

A student has test his airplane and he is far from the airplane for $5$ meter.He start to test his airplane by letting his airplane to move $60$ degree from the horizontal plane with constant velocity for $120$ meter per minute.Find the rate of distance between the student and the plane when the plane is moving 60 degree from the horizontal plane for $10$ meter in the air ?

Answer 1

Using the cosine rule, write the distance $a$ from the observer. Note that $b=5$ and $c=120t$. The angle $\theta$, as seen in the figure is $180^\circ-60^\circ=120^\circ$. You can now calculate $a(t)$. What they ask is the derivative of this quantity with respect to time $t$:$$\frac{d a(t)}{dt}$$

Answer 2

At first we find displacements as triangle side lengths.

Velocities are in proportion to the sides. Dynamic triangles/ parallelogram of displacements, velocities and accelerations all have a similarity.

Absolute velocity of the plane is proportional to $a$ . Velocity of plane P relative to the student S is proportional to $c$.

Ratio of velocity to side length to $ 120/10=12 sec^{-1}$

So the relative velocity $= 12\times 5 \sqrt3= 103.923 \;$ meters/min.