2 (✳)
A point is moving on the unit circle \(\set{ (x, y)\;:\; x^2 + y^2 = 1 }\) in the \(xy\)--plane. At \((2/\sqrt{5}, 1/\sqrt{5})\text{,}\) its \(y\)--coordinate is increasing at rate 3. What is the rate of change of its \(x\)--coordinate?
3 (✳)
The quantities \(P,\ Q\) and \(R\) are functions of time and are related by the equation \(R=PQ\text{.}\) Assume that \(P\) is increasing instantaneously at the rate of \(8\%\) per year and that \(Q\) is decreasing instantaneously at the rate of \(2\%\) per year. That is, \(\dfrac{P'}{P}=0.08\) and \(\dfrac{Q'}{Q}=-0.02\text{.}\) Determine the percentage rate of change for \(R\text{.}\)
4 (✳)
Three quantities, \(F\text{,}\) \(P\) and \(Q\) all depend upon time \(t\) and are related by the equation
\begin{equation*}
F=\frac{P}{Q}
\end{equation*}
- Assume that at a particular moment in time \(P=25\) and \(P\) is increasing at the instantaneous rate of 5 units/min. At the same moment, \(Q=5\) and \(Q\) is increasing at the instantaneous rate of 1 unit/min. What is the instantaneous rate of change in \(F\) at this moment?
- Assume that at another moment in time \(P\) is increasing at the instantaneous rate of \(10\%\) and \(Q\) is decreasing at the instantaneous rate \(5\%\text{.}\) What can you conclude about the rate of change of \(F\) at this moment?
5 (✳)
Two particles move in the Cartesian plane. Particle A travels on the \(x\)-axis starting at \((10,0)\) and moving towards the origin with a speed of \(2\) units per second. Particle B travels on the \(y\)-axis starting at \((0,12)\) and moving towards the origin with a speed of \(3\) units per second. What is the rate of change of the distance between the two particles when particle A reaches the point \((4,0)\text{?}\)
6 (✳)
Two particles \(A\) and \(B\) are placed on the Cartesian plane at \((0,0)\) and \((3,0)\) respectively. At time 0, both start to move in the \(+y\) direction. Particle \(A\) moves at 3 units per second, while \(B\) moves at \(2\) units per second. How fast is the distance between the particles changing when particle \(A\) is at a distance of \(5\) units from \(B\text{.}\)
7 (✳)
Ship A is 400 miles directly south of Hawaii and is sailing south at 20 miles/hour. Ship B is 300 miles directly east of Hawaii and is sailing west at 15 miles/hour. At what rate is the distance between the ships changing?
8 (✳)
Two tall sticks are vertically planted into the ground, separated by a distance of \(30\) cm. We simultaneously put two snails at the base of each stick. The two snails then begin to climb their respective sticks. The first snail is moving with a speed of \(25\) cm per minute, while the second snail is moving with a speed of \(15\) cm per minute. What is the rate of change of the distance between the two snails when the first snail reaches \(100\) cm above the ground?
9 (✳)
A \(20\)m long extension ladder leaning against a wall starts collapsing in on itself at a rate of \(2\)m/s, while the foot of the ladder remains a constant \(5\)m from the wall. How fast is the ladder moving down the wall after \(3.5\) seconds?
10
A watering trough has a cross section shaped like an isosceles trapezoid. The trough is 2 metres long, 50 cm high, 1 metre wide at the top, and 60 cm wide at the bottom.
A pig is drinking water from the trough at a rate of 3 litres per minute. When the height of the water is 25 cm, how fast is the height decreasing?
11
A tank is 5 metres long, and has a trapezoidal cross section with the dimensions shown below.
A hose is filling the tank up at a rate of one litre per second. How fast is the height of the water increasing when the water is 10 centimetres deep?
12
A rocket is blasting off, 2 kilometres away from you. You and the rocket start at the same height. The height of the rocket in kilometres, \(t\) hours after liftoff, is given by
\begin{equation*}
h(t)=61750t^2
\end{equation*}
How fast (in radians per second) is your line of sight rotating to keep looking at the rocket, one minute after liftoff?
13 (✳)
A high speed train is traveling at 2 km/min along a straight track. The train is moving away from a movie camera which is located 0.5 km from the track.
- How fast is the distance between the train and the camera increasing when they are 1.3 km apart?
- Assuming that the camera is always pointed at the train, how fast (in radians per min) is the camera rotating when the train and the camera are 1.3 km apart?
14
A clock has a minute hand that is 10 cm long, and an hour hand that is 5 cm long. Let \(D\) be the distance between the tips of the two hands. How fast is \(D\) decreasing at 4:00?
15 (✳)
Find the rate of change of the area of the annulus \(\{ (x, y) \;:\; r^2 \le x^2 + y^2 \le R^2 \}\text{.}\) (i.e. the points inside the circle of radius \(R\) but outside the circle of radius \(r\)) if \(R = 3 \;\mathrm{cm}\text{,}\) \(r = 1\;\mathrm{cm}\text{,}\) \(\ds\diff{R}{t} = 2\;\frac{\mathrm{cm}}{\mathrm{s}}\text{,}\) and \(\ds\diff{r}{t} = 7\;\frac{\mathrm{cm}}{\mathrm{s}}\text{.}\)
16
Two spheres are centred at the same point. The radius \(R\) of the bigger sphere at time \(t\) is given by \(R(t)=10+2t\text{,}\) while the radius \(r\) of the smaller sphere is given by \(r(t)=6t\text{,}\) \(t \ge 0\text{.}\) How fast is the volume between the spheres (inside the big sphere and outside the small sphere) changing when the bigger sphere has a radius twice as large as the smaller?
17
You attach two sticks together at their ends, and stick the other ends in the mud. One stick is 150 cm long, and the other is 200 cm.
The structure starts out being 1.4 metres high at its peak, but the sticks slide, and the height decreases at a constant rate of three centimetres per minute. How quickly is the area of the triangle (formed by the two sticks and the level ground) changing when the height of the structure is 120 cm?
18
The circular lid of a salt shaker has radius 8. There is a cut-out to allow the salt to pour out of the lid, and a door that rotates around to cover the cut-out. The door is a quarter-circle of radius 7 cm. The cut-out has the shape of a quarter-annulus with outer radius 6 cm and inner radius 1 cm. If the uncovered area of the cut-out is \(A\) cm\(^2\text{,}\) then the salt flows out at \(\frac{1}{5}A\) cm\(^3\) per second.
Recall: an annulus is the set of points inside one circle and outside another, like a flat doughnut (see Question 3.2.2.15).
While pouring out salt, you spin the door around the lid at a constant rate of \(\frac{\pi}{6}\) radians per second, covering more and more of the cut-out. When exactly half of the cut-out is covered, how fast is the flow of salt changing?
19
A cylindrical sewer pipe with radius 1 metre has a vertical rectangular door that slides in front of it to block the flow of water, as shown below. If the uncovered area of the pipe is \(A\) m\(^2\text{,}\) then the flow of water through the pipe is \(\frac{1}{5}A\) cubic metres per second.
The door slides over the pipe, moving vertically at a rate of 1 centimetre per second. How fast is the flow of water changing when the door covers the top 25 centimetres of the pipe?
20
A martini glass is shaped like a cone, with top diameter 10 cm and side length 10 cm.
When the liquid in the glass is 7 cm high, it is evaporating at a rate of 5 mL per minute. How fast is the height of the liquid decreasing?
