Related Rates
1) A spherical snowball melts in such a way that the instant at which its radius is 20
cm, its radius is decreasing at 3 cm/min. At what rate is the volume of the ball of
snow changing at that instant?
2) A spherical snowball is melting. Its radius decreases at a constant rate of 2 cm
per minute from an initial value of 70 cm. How fast is the volume decreasing half
an hour later?
3) A 3 meter ladder stands against a wall. The foot of the ladder moves outward at a
speed of .1 meters/sec. when the foot is 1 meter from the wall. At that moment,
how fast is the top of the ladder falling? What if the foot has been 2 meters from
the wall?
4) An airplane flying at 450 km/hr at a constant altitude of 5 km, is approaching a
camera mounted on the ground. Let θ be the angle of elevation above the
ground at which the camera is pointed. When θ = π / 3 , how fast does the
camera have to rotate in order to keep the plane in view?
Related Rates
5) Two cars start moving from the same point. One travels south at 60 mph and the
other travels west at 25 mph. At what rate is the distance between the cars
increasing two hours later?
6) Gravel is being dumped from a conveyor belt at a rate of 30 ft³/min and its
coarseness is such that it forms a pile in the shape of a cone whose base diameter
and height are always equal. How fast is the height of the pile increasing when
the pile is 10 ft high?
7) Boyle’s Law states that when a sample of gas is compressed at a constant
temperature, the pressure P and volume V satisfy the equation PV = C, where C is
a constant. Suppose that at a certain instant the volume is 600 cm³, the pressure is
150 kPa, and the pressure is increasing at a rate of 20 kPa/min. At what rate is the
volume decreasing at this instant?
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8. A hot-air balloon rising straight up from a level field is tracked by a range finder
π
500 ft from the lift-off point. At the moment the range finder’s elevation angle is ,
4
the angle is increasing at the rate of 0.14 rad/min. How fast is the balloon rising at
that moment?
θ
500 ft
9. Water runs into a conical tank at the rate of 9 ft³/min. The tank stands point down
and has a height of 10ft and a base radius of 5ft. How fast is the water level rising
when the water is 6ft deep?
10. A trough 3ft wide and 12ft long is being filled at a rate of 2 cubic feet per minute. The
ends of the trough are isosceles triangles with altitudes 3ft. How fast is the water level
rising when the depth is 1ft?