1

A weather balloon is rising vertically at the rate of \(10\) ^ft⁄_s. An observer is standing on the ground \(300\) ft horizontally from the point where the balloon was released. At what rate is the distance between the observer and the balloon changing when the balloon is \(400\) ft height?

2

At 2:00 PM one day, Muffin and Rex were sleeping atop one another. A loud noise startled the animals and Muffin began to run due north at a constant rate of \(1.7\) ^ft⁄_s while Rex ran due east at a constant rate of \(2.1\) ^ft⁄_s. The critters maintained these paths and rates for several seconds. Calculate the rate at which the distance between the cat and dog was changing three seconds into their run.

3

Schuyler's clock is kaput; the minute hand functions as it should but the hour hand is stuck at \(4\text{.}\) The minute hand on the clock is \(30\) cm long and the hour hand is \(20\) cm long. Determine the rate of change between the tips of the hands every time the minute hand points directly at \(12\text{.}\)

4

It was a dark night and \(5.5\) ft tall Bahram was walking towards a street lamp whose light was perched \(40\) ft into the air. As he walked, the light caused a shadow to fall behind Bahram. When Bahram was \(80\) ft from the base of the lamp he was walking at a pace of \(2\) ^ft⁄_s. At what rate was the length of Bahram's shadow changing at that instant? Make sure that each stated and calculated rate value has the correct sign. Make sure that your conclusion sentence clearly communicates whether the length of the shadow was increasing or decreasing at the indicated time.

5

The gravitational force, \(F\text{,}\) between two objects in space with masses \(m_1\) and \(m_2\) is given by the formula \(F=\frac{Gm_1m_2}{r^2}\) where \(r\) is the distance between the objects' centers of mass and \(G\) is the universal gravitational constant.

Two pieces of space junk, one with mass \(500\) kg and the other with mass \(3000\) kg, were drifting directly toward one another. When the objects' centers of mass were \(250\) km apart the lighter piece was moving at a rate of \(0.5\) ^km⁄_h and the heavier piece was moving at a rate of \(0.9\) ^km⁄_h. Leaving \(G\) as a constant, determine the rate at which the gravitational force between the two objects was changing at that instant. Make sure that each stated and calculated rate value has the correct sign. Make sure that your conclusion sentence clearly communicates whether the force was increasing or decreasing at that given time. The unit for \(F\) is Newtons (N).

6

An eighteen inch pendulum sitting atop a table is in the downward part of its motion. The pivot point for the pendulum is \(30\) in above the table top. When the pendulum is \(45\) ° away from vertical, the angle formed at the pivot is decreasing at the rate of \(25\) ^°⁄_s. At what rate is the end of the pendulum approaching the table top at this instant? Make sure that each stated and calculated rate value has the correct sign. Make sure that you use appropriate units when defining your variables.