A Collection of Problems in Differential Calculus: Problems from Calculus I Final Examinations, 2000-2020 Department of Mathematics, Simon Fraser University

Section 2.3 Related Rates

To solve a related rates problem you need to do the following:

1. Identify the independent variable on which the other quantities depend and assign it a symbol, such as \(t\text{.}\) Also, assign symbols to the variable quantities that depend on \(t\text{.}\)

2. Find an equation that relates the dependent variables.

3. Differentiate both sides of the equation with respect to \(t\) (using the chain rule if necessary).

4. Substitute the given information into the related rates equation and solve for the unknown rate.

Solve the following related rates problems:

1. A bug is walking on the parabola \(y=x^2\text{.}\) At what point on the parabola are the \(x-\) and \(y-\) coordinates changing at the same rate?

2. A particle is moving along the parabola \(y=x^2-4x+8\text{.}\) Its \(x-\)coordinate as a function of time \(t\) (in seconds) is \(x(t)=-2t^3+5\) metres. Let \(l\) be the line joining the origin \((0,0)\) to the particle. Determine how quickly the angle between the \(x\)-axis and the line \(l\) is changing when \(x=3\text{.}\)

3. A child 1.5 m tall walks towards a lamppost on the level ground at the rate \(0.25\) m/sec. The lamppost is 10 m high. How fast is the length of the child's shadow decreasing when the child is 4 m from the post?

4. A light moving at \(2\) m/sec approaches a 2-m tall man standing 4 m from a wall. The light is 1 m above the ground. How fast is the tip \(P\) of the man's shadow moving up the wall when the light is 8 m from the wall? (See Figure 2.3.1.)

   

5. A ladder 15 ft long rests against a vertical wall. Its top slides down the wall while its bottom moves away along the level ground at a speed of 2 ft/s. How fast is the angle between the top of the ladder and the wall changing when the angle is \(\pi /3\) radians?

6. A ladder 12 metres long leans against a wall. The foot of the ladder is pulled away from the wall at the rate \(\frac{1}{2}\) m/min. At what rate is the top of the ladder falling when the foot of the ladder is 4 metres from the wall?

7. A rocket \(R\) is launched vertically and it is tracked from a radar station \(S\) which is \(4\) miles away from the launch site at the same height above sea level. At a certain instant after launch, \(R\) is 5 miles away from \(S\) and the distance from \(R\) to \(S\) is increasing at a rate of 3600 miles per hour. Compute the vertical speed \(v\) of the rocket at this instant.

8. A boat is pulled into a dock by means of a rope attached to a pulley on the dock, Figure 2.3.2. The rope is attached to the bow of the boat at a point 1 m below the pulley. If the rope is pulled through the pulley at a rate of 1 m/sec, at what rate will the boat be approaching the dock when 10 m of rope is out?

   

9. A ship is moving on the surface of the ocean in a straight line at 10 km/hr. At the same time, an enemy submarine maintains a position directly below the ship while diving at an angle of 30 degrees below the horizontal.

   1. provide a diagram of this problem situation. Use \(x\) as the line of movement of the ship in kilometres and \(y\) as the depth of the submarine in kilometres.

   2. How fast is the submarine depth increasing?

10. A person (\(A\)) situated at the edge of the river observes the passage of a speed boat going downstream. The boat travels exactly through the middle of the river (at the distance \(d\) from the riverbank.) The river is 10 m wide. When the boat is at \(\theta =60^0\) (see Figure 2.3.3) the observer measures the rate of change of the angle \(\theta\) to be 2 radians/second. What is the speed, \(v\text{,}\) of the speed boat at that instant?

    

11. A high speed train is traveling at 3 km/min along a straight track. The train is moving away from a movie camera which is located 0.5 km from the track. The camera keeps turning so as to always point at the front of the train. How fast (in radians per minute) is the camera rotating when the train is 1 km from the camera? See Figure 2.3.4.

    

12. An airplane is flying horizontally at an altitude of \(y=3\) km and at a speed of 480 km/h passes directly above an observer on the ground. How fast is the distance \(D\) from the observer to the airplane increasing 30 seconds later?

13. An airplane flying horizontally at a constant height of \(1000\) m above a fixed radar station. At a certain instant the angle of elevation \(\theta\) from the station is \(\ds \frac{\pi }{4}\) radians and decreasing at a rate of 0.1 rad/sec. What is the speed of the aircraft at this moment.

14. A kite is rising vertically at a constant speed of 2 m/s from a location at ground level which is 8 m away from the person handling the string of the kite. (See Figure 2.3.5.)

    

    1. Let \(z\) be the distance from the kite to the person. Find the rate of change of \(z\) with respect to time \(t\) when \(z=10\text{.}\)

    2. Let \(x\) be the angle the string makes with the horizontal. Find the rate of change of \(x\) with respect to time \(t\) when the kite is \(y=6\) m above ground.

15. A girl flying a kite holds the string 5 feet above the ground level and lets the string out at a rate of 2 ft/sec as the kite moves horizontally at an altitude of 105 feet (see Figure 2.3.6). Assuming there is no sag in the string, find the rate at which the kite is moving when 125 feet of string has been let out.

    

16. A balloon is rising at a constant speed 4m/sec. A boy is cycling along a straight road at a speed of 8m/sec. When he passes under the balloon, it is 36 metres above him. How fast is the distance between the boy and balloon increasing 3 seconds later.

17. A boy is standing on a road holding a balloon and a girl is running towards him at 4 m/sec. At \(t=0\) the girl is 10 m away and the boy releases the ballon which rises vertically at a speed of 2 m/sec. How fast is the distance from the girl to the ballon changing 2 seconds later?

18. A helicopter takes off from a point 80 m away from an observer located on the ground, and rises vertically at 2 m/s. At what rate is the elevation angle of the observer's line of sight to the helicopter changing when the helicopter is 60 m above the ground.

19. An oil slick on a lake is surrounded by a floating circular containment boom. As the boom is pulled in, the circular containment area shrinks (all the while maintaining the shape of a circle.) If the boom is pulled in at the rate of 5 m/min, at what rate is the containment area shrinking when it has a diameter of 100m?

20. A rectangle is inscribed in the unit circle so that its sides are parallel to the coordinate axis. Let \(\theta \in \left(0,\frac{\pi}{2}\right)\) be the angle between the positive \(x\)-axis and the ray with the initial point at the origin and passing through the top-right vertex \(P\) of the rectangle. Suppose that the angle \(\theta\) is increasing at the rate of \(2\mbox{ radians} /\mbox{second}\) and suppose that all lengths are measured in centimetres. At which rate is the area of the rectangle changing when \(\displaystyle \theta =\frac{\pi}{3}\text{?}\) Is the area increasing or decreasing at that moment? Why? Show all your work. Do not forget to use the appropriate units. Clearly explain your reasoning.

21. Consider a cube of variable size. (The edge length is increasing.) Assume that the volume of the cube is increasing at the rate of 10 cm\(^3\)/minute. How fast is the surface area increasing when the edge length is 8 cm?

22. Consider a cube of variable size. (The edge length is increasing.) Assume that the surface area of the cube is increasing at the rate of \(6\) cm\(^2\)/minute. How fast is the volume increasing when the edge length is 5 cm?

23. At what rate is the diagonal of a cube changing if its edges are decreasing at a rate of \(3\) cm/s? (See Figure 2.3.7.)

    

24. The volume of an ice cube is decreasing at a rate of 5 m\(^3\)/s. What is the rate of change of the side length at the instant when the side lengths are 2 m?

25. The height of a rectangular box is increasing at a rate of 2 metres per second while the volume is decreasing at a rate of 5 cubic metres per second. If the base of the box is a square, at what rate is one of the sides of the base decreasing, at the moment when the base area is 64 square metres and the height is 8 metres?

26. A coffee filter has the shape of an inverted cone with a fixed top radius \(R\) and height \(H\text{.}\) Water drains out of the filter at a rate of 10 cm\(^3\)/min. When the depth \(h\) of the water is 8 cm, the depth is decreasing at a rate of 2 cm/min.

    1. Express the volume of the cone as a function of the depth of the water only.

    2. What is the ratio \(R/H\text{?}\)

27. Sand is pouring out of a tube at 1 cubic metre per second. It forms a pile which has the shape of a cone. The height of the cone is equal to the radius of the circle at its base. How fast is the sandpile rising when it is 2 metres high?

28. Gravel is being dumped from a conveyer belt at a rate of 1 cubic metre per second. It makes a pile in the the shape of a cone whose base diameter and height are always equal. How fast is the height of the pile changing when it is 5 metres high?

29. A water tank is in the shape of a cone with its vertex pointing downwards. The tank has a radius of 3 m and is 5 m high. At first the tank is full of water, but at time \(t=0\) (in seconds), a small hole at the vertex is opened and the water begins to drain. When the height of water in the tank has dropped to 3 m, the water is flowing out at 2 m\(^3\)/s. At what rate, in metres per second, is the water level dropping then?

30. A conical tank with an upper radius of 4 m and a height of 5 m drains into a cylindrical tank with radius of 4 m and a height of 5 m. (See Figure 2.3.8.) The water level in the conical tank is dropping at a rate of 0.5 m/min when the water level of the conical tank is 3m. At what rate is the water level in the cylindrical tank rising at that point?

    

31. A boy starts walking north at a speed of 1.5 m/s, and a girl starts walking west from the same point \(P\) at the same time at a speed of 2 m/s. At what rate is the distance between the boy and the girl increasing 6 seconds later?

32. At noon of a certain day, ship \(A\) is 60 miles due north of ship \(B\text{.}\) If ship \(A\) sails east at speed of 15 miles per hour and \(B\) sails north at speed of 12.25 miles per hour, determine how rapidly the distance between them is changing 4 hours later?

33. A police car, approaching a right-angled intersection from the north, is chasing a speeding SUV that has turned the corner and is now moving straight east. When the police car is 0.6 km north of the intersection and the SUV is 0.8 km east of the intersection, the police determine with radar that the distance between them and the SUV is increasing at 20 km/hr. If the police car is moving at 60 km/hr at the instant of measurement, what is the speed of the SUV?

34. A ball lands at a point \(A\text{.}\) As soon as the ball lands, Puppy \(\#1\) starts off 15m North of \(A\) running at 3 m/s in the direction of \(A\text{,}\) and Puppy \(\#2\) starts off 12m East of \(A\) running at 2 m/s in the direction of \(??\text{.}\) At what rate is the distance between the puppies changing when they are \(5\)m apart. In answering this question, sketch a diagram and define your terms clearly.

35. A person is walking east along a river bank path at a speed of \(2m/s\text{.}\) A cyclist is on a path on the opposite bank cycling west at a speed of \(5 m/s\text{.}\) The cyclist is initially 500\(m\) east of the walker. If the paths are \(30m\) apart how fast is the distance between the walker and cyclist changing after one minute? (See Figure 2.3.9.)

    

36. A lighthouse is located on a small island 3 km off-shore from the nearest point \(P\) on a straight shoreline. Its light makes 4 revolutions per minute. How fast is the light beam moving along the shoreline when it is shining on a point 1 km along the shoreline from \(P\text{?}\)

37. You are riding on a ferris wheel of diameter 20 metres. The wheel is rotating at 1 revolution per minute. How fast are you rising when you are at the point \(P\) in Figure 2.3.10, that is you are 6 metres horizontally away from the vertical line passing the centre of the wheel?

    

38. Figure 2.3.11 shows a rotating wheel with radius \(40\) cm and a connecting rod \(AP\) with length \(1.2\) m. The pin \(P\) slides back and forth along the \(x\)-axis as the wheel rotates counterclockwise at a rate of \(360\) revolutions per minute.

    

    1. Find the angular velocity of the connecting rod, \(\frac{d\alpha}{dt}\text{,}\) in radians per second, when \(\theta=\frac{\pi}{3}\text{.}\)

    2. Express the distance \(x=|OP|\) in terms of \(\theta\text{.}\)

    3. Find an expression for the velocity of the pin \(P\) in terms of \(\theta\text{.}\)