Related Rates
1. Rachel is pumping air into a spherical balloon so that its volume increases at a rate of 100cm3/s. How fast is the radius of the balloon increasing when the diameter is 50cm?
2. James spills oil from a tanker off the coast of Spain. The oil spills in a circular pattern and the radius increases at a constant rate of 2m/s. How fast is the area of the spill increasing when the radius of the spill is 60m?
3. Suppose the other James is clearing liquid of sediment by pouring it through a conical filter that is 16cm high and has a radius of 4cm at the top. Suppose also that the liquid flows out of the cone at a constant rate of 2 cm3/min.
a) Do you think that the depth of the liquid will decrease at a constant rate?
b) Show your conclusion is true:
c)At what rate is the depth of the liquid changing at the instant when the level is 8cm deep?
4. Suppose Jeremy escapes calculus class in a hot air balloon and is tracked by envious students 500m away from his launch point. When the angle of elevation is π/4, the angle is increasing at 0.14 radians per minute. How fast is Jeremy rising at that moment?
5. Marissa (running) and Emily (in her car) are both heading south. Marissa, in an effort to lose Emily, turns east. When Marissa is 800m east of the intersection and Emily is 600m north of it, Emily determines with her radar that the distance between her and Marissa is at that point is increasing at 20km/h. If Emily is traveling at 60km/h, how fast is Marissa running?
1. A young child is holding the end of a 20m long ladder. The child runs away for ice cream and the ladder slips down the
wall at a rate of 1m/s. How fast the foot of the ladder slipping away from the house when it is 15m from the house?
(ans = √(7)/3)
2. The volume of a sphere is increasing at a rate of 5m3/min. Find the rate at which the radius is changing
when the diameter is 1.5m. (ans = 20/(9π)m/min.)
3. A cylindrical beaker of radius 10cm is being drained at the rate of 25cm3/min. Find the rate at
which the depth of the liquid is decreasing at the instant the volume of liquid is 1000πcm3 and find
the find the rate at which the depth of the liquid is decreasing when the depth is 2cm.
4. A weather balloon is rising vertically at 2m/sec. An observer is situated 100m from a point on the ground
directly below the balloon. At what rate is the distance between the observer and the balloon changing when
the altitude of the balloon is 500m?
5. When air expands adiabatically (without gaining or losing heat), its pressure P and volume V are related by
the equation PV1.4=C where C is a constant. Suppose that at a certain instant the volume is 400cm3 and the
pressure is 80kPa and is decreasing at a rate of 10kPa/min. At what rate is the volume increasing at this
instant?
6. Two sides of a triangle have lengths 12m and 15m. The angle between them is increasing a t a rate of 2°/min. How
fast is the length of the third side increasing when the angle between the sides of fixed length is 60°?