Download Aim: How do we solve related rates problems

Aim: How do we solve related rates problems?
Objectives: to solve more advanced related rates problem in which students have to set up their own
equation.
Grouping: students are given opportunity to work in cooperative setting during this class. Time is
purposely set aside for students to work collaboratively on at least one set of exercise problems. During
this time, students are encouraged to explain concepts and materials and solutions to each other.
Furthermore, students get to present their solutions on the board.
Differentiated instruction: all students are held to the highest standard. However the students that are
not performing well are grouped with students that are excelling to serve as study partners. Teacher will
informally assess their understanding by asking questions to ensure that they are keeping up.
Assessment: Class is given multiple occasions to work individually and in groups. Teacher circulates
the room to assist and also assess student understanding. Furthermore, students are encouraged to
explain work to each other. This is another opportunity for teacher to assess student understanding.
Lastly, when lesson is finished, students are grouped for class work, questions that are designed as exit
slip problems.
Homework Review (10 minutes): Students are assigned problems to present on board. Students will
also answer questions. Teacher can go over a problem if no one in class could explain it
Lesson Development: The problems we have worked on pertaining to related rates are very simple type
because the equation that relates all quantities is given. In the next two lessons, we will work on
problems in which we have to set up the equation ourselves. Before doing that, we need some
knowledge on velocity. Horizontally (imagine the x-axis), going right constitutes positive velocity and
going left constitutes negative velocity. Vertically (imagine the y-axis), going up means positive
velocity and going down means negative velocity.
The first type of problems will involve right triangles which will break up into Pythagorean Theorem or
a trig equation
EX1 (P185, #4): An airplane flying at an altitude of 6 miles is on a flight path that will take it directly
over a radar tracking station. If the distance between the airplane and the radar station is represented by
s and it is decreasing at a rate of 400 miles per hour when s  10 miles , what is the velocity of the plane
at that moment?
Equation that relates all variables: s 2  36  x 2
d 2
[ s  36  x 2 ]
dt
ds
dx
2s  2 x
dt
dt
ds
2s
dx
dt  10(300)  500 mph

dt
2x
8
Given rate:
ds
 400
dt
Find
dx
when s  10
dt
EX3: An aircraft is flying horizontally at a constant height of 2.75 miles (4000 ft) above a fixed
observation point P. When the airplane is 10 miles away from point P, the speed of the aircraft is 300
mph. How fast is the distance between the aircraft and point P changing at that instant?
dx
 300
dt
2.752  x 2  z 2
dx
dz
 2z
dt
dt
dx
x
dz
102  2.752 (300)
 dt 
 288.433mph
dt
z
10
2x
EX4: An airplane flies at an altitude of 5 miles toward a point directly over an observer. The speed of
the plane is 400 miles per hour.
d
when x = 10 miles.
dt
5
tan  
x
d
5 dx
sec 2 
 2
dt
x dt
d
5 dx
5
102
 2
cos 2  
(400)( 2
)  16 rad/hr
dt
x dt
100
5  102
a) Find
b) How fast is the distance between the plane and the observer changing when x = 10 miles
z 2  52  x 2
dz

dt
x
dx
dt  10(400)  357.771 mph
z
125
Question: how do we know if the right triangle set up translate into a Pythagorean Theorem set up or a
trig equation set up?
EX5: An observer is standing 300 feet from the point at which a balloon is released. The balloon rises at
a rate of 5 feet per second. How fast is the angle of elevation of the observer’s line of sight increasing
when the balloon is 100 feet high?
tan  
y
300
d
1 dy
 cos 2  (
)
dt
300 dt
d
1
300
1

(5)(
)
rad/sec
2
2
dt 300
300  100
2000
HW#28: P188 – 189: 27AC, 31A, 32, 44
HW#28 Solutions: P188 – 189:
27)
c)
31A)
32)
44)
EX1 (P185, #4): An airplane flying at an altitude of 6 miles is on a flight path that will take it directly
over a radar tracking station. If the distance between the airplane and the radar station is represented by
s and it is decreasing at a rate of 400 miles per hour when s  10 miles , what is the velocity of the plane
at that moment?
EX3: An aircraft is flying horizontally at a constant height of 2.75 miles (4000 ft) above a fixed
observation point P. When the airplane is 10 miles away from point P, the speed of the aircraft is 300
mph. How fast is the distance between the aircraft and point P changing at that instant?
EX4: An airplane flies at an altitude of 5 miles toward a point directly over an observer. The speed of
the plane is 400 miles per hour.
a) Find
d
when x = 10 miles.
dt
b) How fast is the distance between the plane and the observer changing when x = 10 miles
EX5: An observer is standing 300 feet from the point at which a balloon is released. The balloon rises at
a rate of 5 feet per second. How fast is the angle of elevation of the observer’s line of sight increasing
when the balloon is 100 feet high?
Aim: How do we solve related rates problems?
EX1 (P185, #4): An airplane flying at an altitude of 6 miles is on a flight path that will take it directly
over a radar tracking station. If the distance between the airplane and the radar station is represented by
s and it is decreasing at a rate of 400 miles per hour when s  10 miles , what is the velocity of the plane
at that moment?
EX2: A 25-foot ladder is leaning against a vertical wall. The floor is slightly slippery and the foot of the
ladder slips away from the wall at the rate of 0.2 inches per second. How fast is the top of the ladder
sliding down the wall when the top is 200 feet above the floor?
EX3: An aircraft is flying horizontally at a constant height of 2.75 miles (4000 ft) above a fixed
observation point P. When the airplane is 10 miles away from point P, the speed of the aircraft is 300
mph. How fast is the distance between the aircraft and point P changing at that instant?
EX4: An airplane flies at an altitude of 5 miles toward a point directly over an observer. The speed of
the plane is 400 miles per hour.
a) Find
d
when x = 10 miles.
dt
b) How fast is the distance between the plane and the observer changing when x = 10 miles
Question: how do we know if the right triangle set up translate into a Pythagorean Theorem set up or a
trig equation set up?
EX5: An observer is standing 300 feet from the point at which a balloon is released. The balloon rises at
a rate of 5 feet per second. How fast is the angle of elevation of the observer’s line of sight increasing
when the balloon is 100 feet high?