3.11 Related RatesMon Dec 1

• Do Now

• Differentiate implicitly in terms of t

• 1)

• 2)

HW Review: p.227 #5-8, 9-27• 5) 17)

• 6) 19)

• 7) 21)

• 8) 23)

• 9) 25)

• 11) 27)

• 13)

• 15)

HW Review: p.227 #10-28• 10) 24)

• 12) book 26)

• 14) 28)

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• 18) 0

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Related Rates• When we use implicit differentiation, we obtain

dy/dx, or the change of y in terms of x.

• In many real life situations, each quantity in an equation changes with time (or another variable)

• In this case, any derivative we find is called a related rate, since each rate in the derivative is related to each other

Related Rates Steps

• 1) Make a simple sketch, if possible• 2) Identify what rate you are looking for• 3) Set up an equation relating ALL of the

relevant quantities• 4) Differentiate both sides of the equation in

terms of the variable you want – if you want dv/dt, you differentiate in terms of t

• 5) Substitute in values we know• 6) Solve for the remaining rate

Ex 1• A 5-meter ladder leans against a wall. The

bottom of the ladder is 1.5 m from the wall at time t=0 and slides away from the wall at a rate of 0.8m/s. Find the velocity of the top of the ladder at time t=1

Ex 2

• Water pours into a fish tank at a rate of 0.3 m^3 / min. How fast is the water level rising if the base of the tank is a rectangle of dimensions 2 x 3 meters?

Ex 3• A spy uses a telescope to track a rocket

launched vertically from a launching pad 6km away. At a certain moment, the angle between the telescope and ground is equal to pi/3 and is changing at a rate of 0.9 radians/min. What is the rocket’s velocity at that moment?

Closure

• Journal Entry: How important is sketching the situation of a related rates problem? How does it help?

• HW: p.199 #1-11 odds, 19-25 odds

• Ch 3 Test Fri

3.11 Related Rates Cont’dTues Dec 2

• Do Now

• Air is being pumped into a spherical balloon at a rate of 5 cm3/min.  Determine the rate at which the radius of the balloon is increasing when the radius of the balloon is 10 cm.

• (hint: Volume = 4/3 pi x r^3)

HW Review p.199 #1-11 19-25 odds

• 1) 0.039 ft/min 21) 1.22 km/min• 3) a) 100pi m^2/min 23) 4.98 rad/hr

b) 24pi m^2/min• 5) 27000pi cm^3/min• 7) 9600pi cm^2/min 25) a) 27.735 km/h• 9) -0.632 m/s b) 112.962 km/h• 11) x = 4.737 m; 0.405 m/s• 19) a) 594.64 km/h b) 0 km/h

More practice

• (Green book) worksheet p.227 #29 31 33 48 51 52

Closure• Hand in: A 15 foot ladder is resting against

the wall.  The bottom is initially x feet away from the wall and is being pushed towards the wall at a rate of 0.5 ft/sec.  How fast is the top of the ladder moving up the wall when the bottom of the ladder is 4 feet from the wall?? (Hint: Use Pythagorean Theorem)

• HW: worksheet p.227 #29 31 33 48 51 52• Ch 3 Test Friday

Do Now• A 15 foot ladder is resting against the wall. 

The bottom is initially x feet away from the wall and is being pushed towards the wall at a rate of 0.5 ft/sec.  How fast is the top of the ladder moving up the wall when the bottom of the ladder is 4 feet from the wall??

HW Review p.227 #29 31 33

• 29) -65 rad/s

• 31) .03 rad/s

• 33) 6pi or 18.85 mm^2/hr

HW Review: p.228 #48 51 52

• 48) 2000pi = 6283 ft^2 / min

• 51) -2.088, -2.332

• 52) 1.59 m/sec