warmup 1) 2). 4.6: related rates they are related (xmas 2013)
TRANSCRIPT
Warmup1)
2)
4.6: Related Rates
They are related (Xmas 2013)
Suppose that the radius of a sphere is changing at a rate of 0.1 cm/sec. The sphere has a radius of 10 cm to startFind the rate of change of the volume.(Possible if the sphere is a soap bubble or a balloon.)
34
3V r
24dV dr
rdt dt
2 cm4 10cm 0.1
sec
dV
dt
3cm
40sec
dV
dt
The sphere is growing at a rate of .340 cm / sec
Air is being pumped into a balloon at a rate of 6 cubic feet per minute.
Find the rate of change of the radius when the radius is 2 ft.
Ex.
Steps for Related Rates Problems:
1. Draw a picture (sketch).
2. Write down known information.
3. Write down what you are looking for.
4. Write an equation to relate the variables.
5. Differentiate both sides with respect to t.
6. Evaluate.
A pebble gets dropped into a pool and the resulting ripplesare concentric circles that expand from the center.The radius of the outer ripple is increasing at a constantrate of 1 foot per second. When the radius is 3 feet, at what rate is the total area of the water changing?
All edges of a cube are expanding at a rate of 4 inches per second. How fast is the volume changing when each edge is (a) 1 inch, (b) 3 inches ?
Hot Air Balloon Problem:
Given:4
rad
0.14min
d
dt
How fast is the balloon rising?
Finddh
dt
tan500
h
2 1sec
500
d dh
dt dt
2
1sec 0.14
4 500
dh
dt
h
500ft
Hot Air Balloon Problem:
Given:4
rad
0.14min
d
dt
How fast is the balloon rising?
Finddh
dt
tan500
h
2 1sec
500
d dh
dt dt
2
1sec 0.14
4 500
dh
dt
h
500ft
2
2 0.14 500dh
dt
1
12
4
sec 24
ft140
min
dh
dt
4x
3y
B
A
5z
Truck Problem:Truck A travels east at 40 mi/hr.Truck B travels north at 30 mi/hr.
How fast is the distance between the trucks changing 6 minutes later?
r t d 1
40 410
130 3
10
2 2 23 4 z 29 16 z
225 z5 z
4x
3y
30dy
dt
40dx
dt
B
A
5z
Truck Problem:
How fast is the distance between the trucks changing 6 minutes later?
r t d 1
40 410
130 3
10
2 2 23 4 z 29 16 z
225 z5 z
2 2 2x y z
2 2 2dx dy dzx y zdt dt dt
4 40 3 30 5dz
dt
250 5dz
dt
50dz
dt
miles50
hour
Truck A travels east at 40 mi/hr.Truck B travels north at 30 mi/hr.
Police chase problem (group):
A police cruiser, approaching a right intersection from the north, is chasing a speeding car that has turned thecorner and headed east. When the cruiser is 0.6 mi north of the intersection and the car is 0.8 mi to the east, the police determine the distance between them is increasing at 20 mph. If the cruiser is moving at 60 mph at the instant measurement,what is the speed of the car?
ex. 3 from our book
Water runs into a conical tank at a rate of 9ft3/min.
The tanks stands point down and has a height of 10ftand a base radius of 5ft. How fast is the water level rising when the water is 6ft deep?
Ex. 4 p. 235
Water is draining from a cylindrical tank at 3 liters/second. How fast is the surface dropping?
L3
sec
dV
dt
3cm3000
sec
Finddh
dt2V r h
2dV dhr
dt dt (r is a constant.)
32cm
3000sec
dhrdt
3
2
cm3000
secdh
dt r
(We need a formula to relate V and h. )
31 L 1000 cm
B) At what rate is the angle changing where theladder meets the ground when the ladder is 10 feet from from the base of the wall.
The end