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Not all equations can be written like y = f(x), so taking the derivative can be tricky. Save the mess and do it directly with 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.

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As shown in the above diagram, a plane taxis off a runway at a speed of \(a \text{ m/s} = 42 \text{ m/s}\) 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?

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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 is \(x=28 \text{ m}\) 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 \(a=18?\)

Note: The above diagram is not drawn to scale.

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