Aim: How do we solve related rate problems?. 3. 5 steps for solving related rate problems Diagram...

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 𝑦𝑑𝑦𝑑𝑥

=4 𝑥 𝑑𝑦𝑑𝑡

=4 𝑥− 1

2 𝑦2.

3. 5 steps for solving related rate problems

DiagramRateEquationDerivativeSubstitution

𝑑𝑟𝑑𝑡

=6 𝐶=2𝜋 𝑟

𝑑𝐶𝑑𝑡

=2𝜋𝑑𝑟𝑑𝑡

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 it’s 9 feet above the ground?

ladder

x

y

9 ft

Given and finding

𝑥2+𝑦2=412 2 𝑥𝑑𝑥𝑑𝑡

+2 y𝑑𝑦𝑑𝑡

=0

+

= 40

= 0 18𝑑𝑦𝑑𝑡

=−320𝑑𝑦𝑑𝑡

=− 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+𝑦2=𝑧 25 is aconstant

2 𝑦𝑑𝑦𝑑𝑡

=2𝑧𝑑𝑧𝑑𝑡

y = 4, =

)𝑑𝑦𝑑𝑡

=4000√418

=500√41

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

y√𝑥2+𝑦2

𝑑𝑥𝑑𝑡

=30 ,𝑑𝑦𝑑𝑡

=60

𝑥=30 ∙13=10 , 𝑦=60 ∙

13=20

𝑃=𝑥+𝑦+√𝑥2+𝑦2

𝑑𝑃𝑑𝑡

= 𝑑𝑥𝑑𝑡

+ 𝑑𝑦𝑑𝑡

+ 12

(𝑥2+𝑦2 )− 12 (2𝑥 𝑑𝑥

𝑑𝑡+2 𝑦 𝑑𝑦

𝑑𝑡)

¿30+60+1

2√102+202∙(2 ∙10 ∙30+2∙ 20 ∙ 60)

¿90+3000

2√500≈ 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

𝑑𝑧𝑑𝑡

=2 𝑥2+72=𝑧 2

=

𝑥=20 , 𝑧=√ 449≈ 21.2

𝑑𝑥𝑑𝑡

=𝑧𝑑𝑧𝑑𝑡𝑥¿

21.2 ∙220

¿42.420

=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 km𝑑𝑎𝑑𝑡

=35 ,𝑑𝑏𝑑𝑡

=25b

a

Let distance = c

𝑐2=1002+(𝑎+𝑏)2

)100 km

)

)

260𝑑𝑐𝑑𝑡

=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

𝐴=12

h𝑏 𝑑𝐴𝑑𝑡

=12𝑏

h𝑑𝑑𝑡

+12

h𝑑𝑏𝑑𝑡

2=12

∙ 20 (− 1 )+ 12

∙ 10𝑑𝑏𝑑𝑡

100=12𝑏 ∙10 ,b = 10

2=−10+5𝑑𝑏𝑑𝑡

𝑑𝑏𝑑𝑡

=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

𝑑𝑥𝑑𝑡

=5 light

z = x + y

𝑦𝑥

=𝑥+𝑦20

10 𝑦=3 𝑥+3 𝑦7 𝑦=3 𝑥

7𝑑𝑦𝑑𝑡

=3𝑑𝑥𝑑𝑡 7

𝑑𝑦𝑑𝑡

=3(5)𝒅𝒚𝒅𝒕

=𝟏𝟓𝟕

𝑑𝑧𝑑𝑡

=𝑑𝑥𝑑𝑡

+𝑑𝑦𝑑𝑡

¿5+157

=𝟓𝟎𝟕

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?

A

BC

𝑑𝐴𝑑𝑡

=40 ,𝑑𝐵𝑑𝑡

=30

𝐴2+𝐵2=𝐶2

2 A𝑑𝐴𝑑𝑡

+2𝐵𝑑𝐵𝑑𝑡

=2𝐶𝑑𝐶𝑑𝑡 = 5

2 (4 ) (40 )+2 (3 ) (30 )=2(5)𝑑𝐶𝑑𝑡

10 = 50 mph