2019 Related Rates
AB Calculus
Intro:
( )( )
x f ty g t
2 3y x
GOAL: to find the rates of change of two (or more) variables with respect to a third variable (the parameter)
This is a adaptation of IMPLICIT functions x and y are implicit functions of t .
ILLUSTRATION:
A point is moving along the parabola,
Find the rate of change of y when x = 1if x is changing at 2 units per second.
moving
moving time
graphdy
ππ¦ππ‘=? ππ₯
ππ‘ =+2 hπ€ πππ₯=1π¦=π₯2+3ππ¦ππ‘=2π₯ ππ₯ππ‘ππ¦
ππ‘=2 (1 ) (2 )=4
PROCEDURE:
1). DRAW A PICTURE! β Determine what rates are being compared.
2). Assign variables to all given and unknown quantities and rates.
3). Write an equation involving the variables whose rates are given or are to be found Β· Equation of a graph?
Β· Formula from Geometry? The equation must involve only the variables from step 2. β
((You may have to solve a secondary equation to eliminate a variable.))
4). Use Implicit Differentiation (with respect to the parameter t).
5). AFTER DIFFERENTIATION, substitute in all known values (( You may have to solve a secondary equation
to find the value of a variable.))
May plug in a constant as long as it is unchanging
Geometry formulas:
Sphere:
Cylinder:
Cone:
Pythagorean Theorem:
343
V r
24SA r
2V r h2LA rh
22 2SA r rh
213
V r h
2 2 2x y z
METHOD: Inflating a Balloon - 1A spherical balloon is inflated so that the radius is changing at a rate of 3 cm/sec. How fast is the volume changing when the radius is 5 cm.?
Draw and label a picture.
List the rates and variables.
Find an equation that relates the variables and rates. (Extra Variables?)
Differentiate (with respect to t.)
Plug in and solve.
Step 1:
ππππ‘ =+3
ππππ‘ =?
When r =5
π=43 π π3
πππ π‘ =4π (52)(3)
πππ π‘ =4π (π 2)
ππππ‘
πππ π‘ =300π ππ3
π ππ
Plugin 5 gives vol not rate of change
Ex 2: Ladder w/ secondary equationA 25 ft. ladder is leaning against a vertical wall. If the bottom of the ladder is pulled horizontally away from the wall at a rate of 3 ft./sec., how fast is the top of the ladder sliding down the wall when the bottom is 15 ft. from the wall?
π§=25ππ₯ππ‘ =+3
ππ¦ππ‘=?
hπ€ πππ₯=15π₯2+ π¦2=π§2
2 π₯ ππ₯ππ‘ +2 π¦ ππ¦ππ‘ =2 π§ ππ§ππ‘
π₯2+ π¦2=π§2
π₯2+ π¦2=252
2 π₯ ππ₯ππ‘ +2 π¦ ππ¦ππ‘ =0
2 (15 ) (3 )+2 (20 ) ππ¦ππ‘ =0
ππ¦ππ‘=
β 4520 =
β 94
90+40 ππ¦ππ‘ =0
The ladder is coming down -2.25 ft/sec
2 (15 ) 3+2 (20 ) ππ¦ππ‘ =0
90+40 ππ¦ππ‘ =0
ππ¦ππ‘=
β 4520 =
β 94
# constant does not change ever you can plug in the equation
The ladder is coming down
152+ π¦2=252
II: Similar Triangles
Similar TrianglesA
B
C
A
B
C
A B
C
D E
F
ABC DEF AB BC CADE EF FD
Similar Triangles may be the whole set up.
Similar Triangles may be required to to eliminate an extra variable β or- to find a missing value
Ex 4:A person is pushing a box up a 20 ft. ramp with a 5 ft. incline at a rate of 3 ft.per sec.. How fast is the box rising?
derivative
zx
y
20 ft
5
π π§ππ‘ =+3
π¦π§=
520
as
5 π§=20 π¦5 ππ§ππ‘ =20 ππ¦ππ‘
20 ππ¦ππ§=5(3)
ππ¦ππ‘=
1520 =
34
ππ‘π ππ
Ex 5:Pat is walking at a rate of 5 ft. per sec. toward a street light whose lamp is 20 ft. above the base of the light. If Pat is 6 ft. tall, determine the rate of change of the length of Patβs shadow at the moment Pat is 24 ft. from the base of the lamppost.
ππ₯ππ‘ ββ 5
ππ¦ππ‘=? hπ€ πππ₯=24
20π₯+π¦=
6π¦ 20 π¦=6 π₯+6 π¦20 ππ¦ππ‘ =6 ππ₯ππ‘ +6 ππ¦ππ‘
14 ππ¦ππ‘ =β30
ππ¦ππ‘=
β 3014 =
β157
How fast is the tip of Patβs shadow changing
The distance of top of shadow from post
6x
y
20
66
y-xy
20π¦ =
6π¦βπ₯
Getting smaller
Ex 6: Cone w/ extra equation
Water is being poured into a conical paper cup at a rate of
cubic inches per second. If the cup is 6 in. tall and the top of the cup has a radius of 2 in., how fast is the water level rising when the water is 4 in. deep?
23
π=13 ππ2 h
Three variables
πππ π‘ =
+23
π hππ‘=? hπ€ ππ h=4
26=
πh
2 h=6πh3=
2 h6 =π
r changesh changes
π=13π( h
3 )2
h
π=13 ππ 2 h
π=π27 h3
πππ π‘ =
π9 h2 hπ
ππ‘23=
π9 42 hπ
ππ‘+3
16π β 23=
hπππ‘
18π=
hπππ‘
Too many variables need to find r
Only two variables
III: Angle of Elevation
hyp
opp
adj
sin π=πππhπ¦π
csc π= hπ¦ππππ
sπππ=hπ¦ππππ
cot π= ππππππ
sin π=πππhπ¦π
cosπ= πππhπ¦π
tanπ=ππππππ
Angles of Elevation
ΞΈa
b
c SOH β CAH - TOA
Hint: The problem may not require solving for an angle measure β¦ only a specific trig ratio.
ie. need sec ΞΈ instead of ΞΈ
ΞΈ3
4
5 π=?
secπ=54
Ex 7:A balloon rises at a rate of 10 ft/sec from a point on the ground 100 ft from an observer. Find the rate of change of the angle of elevation of the balloon from the observer when the balloon is 100 ft. above the ground.
100
100
100 β2When y =100
secπ=100 β2100
=β2
ππ¦ππ‘=40
ππππ‘ =? hπ€ πππ¦=100
tanπ= π¦π₯
orπ¦
100
π ππ2π π π§ππ‘ =1
100ππ¦ππ‘
(β2 )2 π π§ππ‘ =1
100(10 )
2 ππππ‘ =1
10ππππ‘ =
120
ππππ ππ
Ex 8:A fishing line is being reeled in at a rate of 1 ft/sec from a bridge 15 ft above the water. At what rate is the angle between the line and the water changing when 25 ft of line is out.
π π§ππ‘ =β1 ππ‘
π ππππππ‘ =? When z = 25 ft
sin π=15π§ β cscπ= π§
15π§ sin π=15
π§ ()
Ex 9:A television camera at ground level is filming the lift off of a space shuttle that is rising vertically according to the position function
, where s is measured in feet and t in seconds. The camera is is 2000 ft. from the launch pad. Find the rate of change of tin the angle of elevation of the camera 10 sec. after lift off.
250s t
IV: Using multiple rates
Ex 11:If one leg, AB, of a right triangle increases at a rate of 2 in/sec while the other leg, AC, decreases at 3 in/sec, find how fast the hypotenuse is changing when AB is 72 in. and AC is 96 in.
AC
B
Ex 12:A metal rod has the shape of a right circular cylinder. As it is being heated, its length is increasing at a rate of 0.005 cm/min and its diameter is increasing at 0.002 cm/min. At what rate is the volume changing when the rod has a length 40 cm and diameter 3 cm.?
5: AP Questions
Example 12: AP Type
At 8 a.m. a ship is sailing due north at 24 knots(nautical miles per hour) is a point P. At 10 a.m. a second ship sailing due east at 32 knots is a P. At what rate is the distance between the two ships changing at (a) 9 a.m. and (b) 11 a.m.?
Ex 13: AP TypeA right triangle has height 7 cm and the hypotenuse is increasing at a rate of 2 cm/sec. When the hypotenuse is 25 cm, find:
a). the rate of change of the base.
b). The rate of change of the acute angle at the base,
c). The rate of change of the area of the triangle.
AP QuestionA coffee pot is made up of a conical
filter that is 6 in. tall and has a diameter
of 6 in. and a cylindrical pot that is 6 in.
in diameter.
Coffee is draining from the filter into the coffeepot at a rate of 10 in3/sec.
a). How fast is the level in the pot rising when the coffee in the
cone is 5 in. deep?
b). How fast is the level in the cone falling at that instant?
AP Question: Combined Example
At noon a sailboat is 20 km south of a freighter. The sailboat is traveling east at 20 km per hour, and the freighter is traveling south at 40 km per hour.
When is the distance between the boats a minimum?
If the visibility is 10 kilometers, could the people see each other?
At what rate is the distance between the two boats changing at that instant?
Last Update
β’ 11/12/11
β’ BC: