Calculus
# Implicit Differentiation

In a fairy tale, a wizard rides a cloud which is moving to the right at a speed of $15\text{ m/s.}$ The wizard throws a ball vertically upward with a speed of $3\text{ m/s}$ relative to the cloud. When the cloud has moved to the right $30 \text{ m}$ following the throw, what is the velocity of the ball as seen by people on the ground?

Assume that air resistance is negligible and ignore the effect of gravity.

$a \text{ m/s} = 42 \text{ m/s}$ in order to take off. After taking off, the plane continues flying at $42 \text{ m/s}$, with vertical speed $b \text{ m/s} = 24 \text{ m/s}.$ What is the horizontal speed of the plane $5$ seconds after take-off?

As shown in the above diagram, a plane taxis off a runway at a speed of$x=28 \text{ m}$ on each side and the length of the base of the roof is $2a.$ If the collapse of the roof is such that $a$ increases at a rate of $5$ meters per hour, how fast is the height $b$ of the roof collapsing when $a=18?$

As shown in the above digram, a thick layer of snow accumulates on a vertically symmetrical peaked roof, which causes the roof to slowly collapse downward under the weight of the snow. The length of the peaked roof isNote: The above diagram is not drawn to scale.