2.3 Related Rates
1. 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 π/3 radians?
2. A ladder 12 meters long leans against a wall. The foot of the ladder is pulled away from the wall at the rate 12 m/min. At what rate is the top of the ladder falling when the foot of the ladder is 4 meters from the wall?
3. A rocketRis launched vertically and its 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.
4. A boat is pulled into a dock by means of a rope attached to a pulley on the dock, Figure 2.1. 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.
Figure 2.1: Boat, Pulley, and Dock
5. 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 θ = 600 (see figure) the observer measures the rate of change of the angle θ to be 2 radians/second.
? 6
w= 10 m
s
A d
y
"
""
"
"
θ z
sv-
What is the speed, v, of the speed boat at that instant?
6. An airplane 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 Dfrom the observer to the airplane increasing 30 seconds later?
7. An airplane flying horizontally at a constant height of 1000 m above a fixed radar station. At a certain instant the angle of elevation θ at the station is
π
4 radians and decreasing at a rate of 0.1 rad/sec. What is the speed of the aircraft at this moment.
8. 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.
y z
8m kite
x
(a) Letz be the distance from the kite to the person. Find the rate of change of z with respect to timet when z = 10.
(b) 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.
2.3. RELATED RATES 27 9. 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.
10. 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 elevation angle of the observer’s line of sight to the helicopter changing when the helicopter is 60 m above the ground.
11. An oil slick on a lake is surrounded by a floating circular containment boom.
As the boom is pulled in, the 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?
12. 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 cm3/minute. How fast is the surface area increasing when the edge length is 8 cm?
13. The height of a rectangular box is increasing at a rate of 2 meters per second while the volume is decreasing at a rate of 5 cubic meters 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 meters and the height is 8 meters?
14. Sand is pouring out of a tube at 1 cubic meter 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 meters high?
15. A water tank is in the shape of a cone with vertical axis and vertex downward.
The tank has radius 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 m3/s. At what rate, in meters per second, is the water level dropping then?
16. A boy starts walking north at a speed of 1.5 m/s, and a girl starts walking west at 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?
17. At noon of a certain day, the ship A is 60 miles due north of the ship B. If the ship A sails east at speed of 15 miles per hour andB sails north at speed
of 12.25 miles per hour, determine how rapidly the distance between them is changing 4 hours later?
18. A lighthouse is located on a small island three (3) km off-shore from the nearest pointPon a straight shoreline. Its light makes four (4) revolutions per minute.
How fast is the light beam moving along the shoreline when it is shining on a point one (1) km along the shoreline from P?
19. A police car, approaching 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 intersection and the SUV is 0.8 km east of 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?