Ajet ascends at a 10 degree angle from the horizontal with an airspeed of 550 mi/hr (its speed along its line of flight is 550 mi/hr). how fast is the altitude of the jet increasing? if the sun is directly overhead, how fast is the shadow of the jet moving on the ground?

1) Consider the departure point is A (0,0).
After one hour flight with an angle of elevation of 10°, it travelled a distance of 550 mi. (speed 550 mi/h) and reached an altitude of H.
(Imagine a right triangle:   Departure point A (0,0), with angle of    elevation A= 10° and after one hour reaching a point B, at an altitude  H, note that AB = hypotenuse-)

Let's calculate H (after 1 hour/flight)

sin10° = H/AB = H/550 and H= 95.5065 mi, Speed of H = 95.506 mi/h

2) The shadow is represented by the horizontal side, adjacent to 10°.

So when the plane is advancing at 550 mi/h, the shadow is stretched at a speed S.
Now let's calculate the length of the shadow (always after one hour flight)
Pythagoras: 550² =95.5065² +(length of shadow)²
And Length of shadow = 541.65 mi
S= 541.65/1 hour = 541.65 mi/h