Aim: How do we solve related rate problems? Do Now: Differentiate implicitly: 1. x + = 2 2. C = 2 with respect to time (t) 3. A pebble is dropped into a pool of water, generating circular ripples. The radius of the largest ripple is increasing at a constant rate of 6 inches per second. What is the
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Aim: How do we solve related rate problems?. 3. 5 steps for solving related rate problems Diagram Rate Equation Derivative Substitution.
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Aim: How do we solve related rate problems?
Do Now: Differentiate implicitly: 1. x + = 2 2. C = 2 with respect to time (t)
3. A pebble is dropped into a pool of water, generating circular ripples. The radius of the largest ripple is increasing at a constant rate of 6 inches per second. What is the increasing in the circumference of the ripple after 3 seconds have passed?
1.1+2 π¦ππ¦ππ₯
=4 π₯ ππ¦ππ‘
=4 π₯β 1
2 π¦2.
3. 5 steps for solving related rate problems
DiagramRateEquationDerivativeSubstitution
ππππ‘
=6 πΆ=2π π
ππΆππ‘
=2πππππ‘
in/sec
Types of related rate problems
1. Use Pythagorean Theorem
2. Use basic area formula such as triangle, circle, trapezoid or circumference
3. Use similar triangle
4. Use the formulas of volume such as cylinder, cone, sphere and some complicated shape
A 41 ft. ladder is leaning against a wall and the top of it is sliding down the wall while the bottom of it slides away from the wall at 4 ft/sec. How fast is the top sliding when itβs 9 feet above the ground?
ladder
x
y
9 ft
Given and finding
π₯2+π¦2=412 2 π₯ππ₯ππ‘
+2 yππ¦ππ‘
=0
+
= 40
= 0 18ππ¦ππ‘
=β320ππ¦ππ‘
=β 320
18
A rocket, rising vertically, is tracked by a radar station that is on the ground 5 miles from the launchpad. How fast is the rocket rising when it is 4 miles high and its distance from radar station is increasing at a rate of 2000 miles/hour?
station
5
yz
Given finding
52+π¦2=π§ 25 is aconstant
2 π¦ππ¦ππ‘
=2π§ππ§ππ‘
y = 4, =
)ππ¦ππ‘
=4000β418
=500β41
Two cars leave an intersection at the same time, one headed east and the other north. The eastbound car is moving at 30 mph while the northbound car is moving at 60 mph. twenty minutes later, what is the rate of change in the perimeter of the right triangle created using the 2 cars and the intersection.
x
yβπ₯2+π¦2
ππ₯ππ‘
=30 ,ππ¦ππ‘
=60
π₯=30 β13=10 , π¦=60 β
13=20
π=π₯+π¦+βπ₯2+π¦2
ππππ‘
= ππ₯ππ‘
+ ππ¦ππ‘
+ 12
(π₯2+π¦2 )β 12 (2π₯ ππ₯
ππ‘+2 π¦ ππ¦
ππ‘)
ΒΏ30+60+1
2β102+202β(2 β10 β30+2β 20 β 60)
ΒΏ90+3000
2β500β 157.08
A boat is being pulled to a dock at the rate of 2 ft/sec. The pulley is 8 feet from the water and the rope is tied to the boat at 1 feet above the water. How fast is the boat approaching the dock when it is 20 feet away?
1
87
x
z
ππ§ππ‘
=2 π₯2+72=π§ 2
=
π₯=20 , π§=β 449β 21.2
ππ₯ππ‘
=π§ππ§ππ‘π₯ΒΏ
21.2 β220
ΒΏ42.420
=2.12
At noon ship A is 100 km west of ship B. Ship A is sailing south at 35km/hr and ship B sailing north at 25km/hr. How fast is the distance between the ships changing at 4 pm?
AB
100 kmππππ‘
=35 ,ππππ‘
=25b
a
Let distance = c
π2=1002+(π+π)2
)100 km
)
)
260ππππ‘
=14400
km/hr
The height of a triangle is decreasing at a rate of 1 cm/min while the area of the triangle is increasing at a rate of 2 /min. At what rate is the base of the triangle changing when the height is 10 cm and the area is 100 ?
Given finding
π΄=12
hπ ππ΄ππ‘
=12π
hπππ‘
+12
hππππ‘
2=12
β 20 (β 1 )+ 12
β 10ππππ‘
100=12π β10 ,b = 10
2=β10+5ππππ‘
ππππ‘
=125
A 6-foot-tall man walks away from a 20-foot-street light at a rate of 5 ft/sec. At what rate is the tip of his shadow moving when he is 24 feet from the lightpost and at what rate is the length of his shadow increasing?
20
6
x y
ππ₯ππ‘
=5 light
z = x + y
π¦π₯
=π₯+π¦20
10 π¦=3 π₯+3 π¦7 π¦=3 π₯
7ππ¦ππ‘
=3ππ₯ππ‘ 7
ππ¦ππ‘
=3(5)π ππ π
=πππ
ππ§ππ‘
=ππ₯ππ‘
+ππ¦ππ‘
ΒΏ5+157
=πππ
Part B
Part A
Two trucks leave a depot at the same time, Truck A travels east at 40 mph and truck B travels north at 30 mph. How fast is the distance between the trucks changing 6 minutes later when A is 4 miles from the depot and B is 3 miles from the depot?