Mrs. Nguyen – Honors Algebra II – Chapter 2.5, 6.1 – 6.3 Notes – Page 1 NOTES 2.5, 6.1 – 6.3 Name:______________________________ Date:________________Period:_________ Mrs. Nguyen’s Initial:_________________ LESSON 2.5 – MODELING VARIATION Direct Variation y mx b when 0 b or y mx or y kx y kx and 0 k - y varies directly as x - y is directly proportional to x - k is the constant of variation - k is the constant of proportionality Express the statement as an equation. Use the given information to find the constant of proportionality. B is directly proportional to L. If L = 15, then B = 1350. Direct variation as th n power n y kx and 0 k - y varies directly as the th n power of x - y is directly proportional to the th n power of x B is directly proportional to the square of L. If L = 15, then B = 1350. Inverse variation k y or k xy x As , x goes up y goes down or As , y goes up x goes down n varies inversely as f. If n = 3, then f = 5.
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NOTES 2.5, 6.1 – 6.3 Name: L 2.5 MODELING VARIATION · Mrs. Nguyen – Honors Algebra II – Chapter 2.5, 6.1 – 6.3 Notes – Page 4 Radian Measure One radian is the measure of
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LESSON 2.5 – MODELING VARIATION Direct Variation y mx b when 0b or
y mx or y kx y kx and 0k
- y varies directly as x - y is directly proportional to x - k is the constant of variation - k is the constant of proportionality
Express the statement as an equation. Use the given information to find the constant of proportionality. B is directly proportional to L. If L = 15, then B = 1350.
Direct variation as thn power
ny kx and 0k
- y varies directly as the thn power of x - y is directly proportional to the thn power
of x
B is directly proportional to the square of L. If L = 15, then B = 1350.
Inverse variation
ky or k xy
x
As ,x goes up y goes down or As ,y goes up x goes down
Joint variation z kxy z varies directly as x and y
M varies jointly as h and n. If h = 7 and n = 9, then M = 504.
Practice Problems: 1. The pressure P of a sample of gas is directly proportional to the temperature T and inversely proportional to the volume V. a. Write an equation that expresses this variation. b. Find the constant of proportionality if 100L of gas exerts a pressure 33.2 kPa at a temperature of 400 K. c. If the temperature is increased to 500K and the volume is decreased to 80L, what is the pressure of the gas?
2. The power P (measured in horse power, hp) needed to propel a boat is directly proportional to the cube of its speed s. An 80-hp engine is needed to propel a certain boat at 10 knots. Find the power needed to drive the boat at 15 knots.
Practice Problems: The measure of an angle in standard position is given. Find two positive angles and two negative angles that are co-terminal with the given angle. Sketch the angles. 1. 210
angle that intercepts an arc s equal in length to the radius of a circle.
C = 2r 6.28r
The radian measure of an angle of one full revolution is 2 . Since one full circle has 360 ,
Note: In a full revolution, the arc length s is equal to
2C r s . Also, there are just over six radius lengths in a full circle. Therefore, the central
angle is s
r where
is measured in radians.
360 2 rad 180 rad 90
2
rad 60
3
rad
Degrees to Radians
Multiply by 180
Example:
Radians to Degrees Multiply by
180
Example:
Practice Problems: Convert the following angles from degrees to radians and from radians to degrees without using a calculator. 4. 150
5. 7
6
6. 240 7.
11
30
Practice Problems: Convert the following angles from degrees to radians and from radians to degrees using a calculator and round to 3 decimal places. 8. 87.4
Practice Problems: The measure of an angle in standard position is given. Find two positive angles and two negative angles that are co-terminal with the given angle. Sketch the angles.
12. 13
6
13. 3
4
14.
2
3
Length of a Circular Arc
In a circle of radius r, the length s of an arc that subtends a central angle of radians is: s r Note: must be in radians.
Example:
Review Problem 15: Find the following arc lengths using geometry then use s r to validate your answers. Given 24CB in ,
0 60m A B , find the following arc lengths using 2 methods
Practice Problems: Find the unknown value. 16. A central angle in a circle of radius 24 cm is subtended by an arc of length 6 cm. Find the measure of in radians.
17. Find the radius of the circle if an arc of length
8 in on the circle subtends a central angle of 4
.
18. A bicycle’s wheels are 14 inches in diameter. How far (in miles) will the bike travel if its wheels revolve 500 times without slipping?
19. How many revolutions will a Ferris wheel of diameter 60 feet make as the Ferris wheel travels a distance of a ½ mile?
20. An ant is sitting 5 cm from the center of a c.d.. If the c.d. turns 40 , how far has the ant moved in meters?
21. A bug is on a car’s windshield wiper and is 10 inches from the base of the windshield wiper. If the bug moves 34 inches, at what angle did the windshield wiper turn?
Angular Speed The angular velocity of a point on a rotating object is the
number of degrees (radians, revolutions, etc.) per unit time through which the point turns.
t
Linear Speed The linear velocity of a point on a rotating object is the distance per unit time that the point travels along its circular path.
sv
t
Note: The linear velocity depends on how far the object is from the axis of rotation, whereas the angular velocity is the same no matter where the object lies on the rotating object.
Relationship between Linear and Angular Speed
If a point moves along a circle of radius r with angular speed , then its linear speed v is given by: v r
Practice Problems: Solve the following problems. 22. A woman is riding a bike whose wheels are 26 inches in diameter. If the wheels rotate at 125 revolutions per minute (rpm), find the speed at which she is traveling, in miles per hour.
23. The rear wheels of a tractor are 4 feet in diameter, and turn at 20 rpm. (a) How fast is the tractor going (feet per second)? (b) The front wheels have a diameter of only 1.8 feet. What is the linear velocity of a point on their tire treads? (c) What is the angular velocity of the front wheels in rpm?
24. The pedals on a bike turn the front sprocket at 8 radians per second. The sprocket has a diameter of 20 cm. The back sprocket, connected to the wheel, has a diameter of 6 cm. (a) Find the linear velocity of the chain. (b) Find the angular velocity of the back sprocket.
25. Dan and Ella are riding on a Ferris wheel. Dan observes that it takes 20 seconds to make a complete revolution. Their seat is 25 feet from the axle of the wheel.
(a) What is their angular velocity? (b) What is their linear velocity?
Practice Problems: Evaluate without using a calculator. Draw and label triangles. 3. tan 60
4. csc45
5. tan30
6. sec30
Practice Problems: Evaluate the given expression without using a calculator. Leave your answer in simplest radical form. 7. sin 60 cos30 8. tan 45 cot(60)
The angle of elevation is the angle from a horizontal line UP to an object. The angle of depression is the angle from a horizontal line DOWN to an object.
Practice Problems: Set up an equation for each word problem and solve. 14. Suppose you have been assigned the job of measuring the height of the local water tower. Climbing makes you dizzy so you decide to do the whole job at ground level. From a point 47.3 meters from the base, you find that you must look up at an angle of 53 degrees to see the top of the tower. How high is the tower?
15. When landing, a jet will average a 3 angle of descent. What is the altitude, to the nearest foot, of a jet on final descent as it passes over an airport radar 6 miles from the start of the runway?
16. At a point 300 feet from the base of a building, the angle of elevation to bottom of a smokestack is 40 , and the angle of elevation to the top is 55 . Find the height of the smokestack alone.
17. The distance between a plane and a building on the ground is 350 feet. The angle of depression from the plane to the building is 20 . Find the horizontal distance from the plane to the building.
Practice Problem 6: Find the reference angle ' for the following. Graph the angle a. 309 '
b. 145 '
c. 7
'4
d. 11
'3
Evaluating Trigonometric Functions of any Angle
To find the value of a trig function of any . 1. Determine the function value for the associated ' . 2. Depending on the quadrant in which lies, affix the appropriate sign to the
function value.
Let be an angle in standard position. Its reference angle is the acute angle ' formed by the terminal side of and the x-axis.