Quiz 4-3Quiz 4-3Factor:Factor:

1. 1. 16)( 2 xxf

2. 2. 34)( 2 xxxf

Quadratic word Quadratic word problemsproblems 02 cbxx

4-9B4-9B

Real WorldReal WorldWhat path does What path does the cannon ball take?the cannon ball take?

Any object in freefall and Any object in freefall and subject to gravitational fieldsubject to gravitational field travels in a parabolic path.travels in a parabolic path.

cbxaxy 2

Projectile MotionProjectile Motion

1005016)( 2 ttts

A A verticalvertical time-distance problem in time-distance problem in twotwo dimensions. dimensions.

Height as a Height as a function of time.function of time.

Height at time = 0Height at time = 0(initial height)(initial height)

Vertical component of velocity Vertical component of velocity (speed) multiplied by time gives the (speed) multiplied by time gives the change in vertical change in vertical position resulting from the initial position resulting from the initial velocity.velocity.

Vertical componentVertical componentof acceleration (ofof acceleration (ofGravity) multiplied Gravity) multiplied by time squaredby time squaredgives the change in gives the change in vertical position due vertical position due to gravity.to gravity.

The projectile motion equation forThe projectile motion equation for units in feet and seconds. units in feet and seconds.

00216)( stvtts Height at time = 0Height at time = 0

(initial height)(initial height)

Vertical component of velocityVertical component of velocityVertical componentVertical componentof accelerationof acceleration

Don’t worry, I’ll give you the equation Don’t worry, I’ll give you the equation with the with the numbers in it. In Pre-Calculus and numbers in it. In Pre-Calculus and math 1050math 1050 you’ll have to come up with the you’ll have to come up with the equation on yourequation on your own.own.

Projectile Motion ProblemProjectile Motion ProblemAn object is launched An object is launched verticallyvertically upward from the ground at upward from the ground at

an initial velocity of 250 ft per second. an initial velocity of 250 ft per second.

c. When will the object fall back down to the ground?c. When will the object fall back down to the ground?

ttts 25016)( 2

a.a. When will the object reach its maximum height?When will the object reach its maximum height?

b. What is the object’s maximum height?b. What is the object’s maximum height?

Projectile Motion ProblemProjectile Motion ProblemAn object is launched An object is launched verticallyvertically upward from the ground at upward from the ground at

an initial velocity of 250 ft per second. an initial velocity of 250 ft per second.

ttts 25016)( 2 a.a. When will the object reach its maximum height?When will the object reach its maximum height?

Projectile Motion ProblemProjectile Motion ProblemAn object is launched An object is launched verticallyvertically upward from the ground at upward from the ground at

an initial velocity of 250 ft per second. an initial velocity of 250 ft per second.

ttts 25016)( 2 a.a. WhenWhen will the object reach its maximum height? will the object reach its maximum height?

(7.8, 976.6): Which is time? Which is height?(7.8, 976.6): Which is time? Which is height?

b. What b. What is its maximum height? is its maximum height?

Projectile Motion ProblemProjectile Motion ProblemAn object is launched An object is launched verticallyvertically upward from the ground at upward from the ground at

an initial velocity of 250 ft per second. an initial velocity of 250 ft per second.

ttts 25016)( 2 c. Whenc. When will the object fall back to the ground? will the object fall back to the ground?

(15.6, 0) Which is time? Which is height?(15.6, 0) Which is time? Which is height?

Projectile Motion ProblemProjectile Motion ProblemAn object is launched An object is launched verticallyvertically upward from the top of a upward from the top of a

100 foot building at an initial velocity of 175 ft per second. 100 foot building at an initial velocity of 175 ft per second.

c. When will the object fall back down to the ground?c. When will the object fall back down to the ground?

10017516)( 2 ttts

a.a. When will the object reach its maximum height?When will the object reach its maximum height?

b. What is the object’s maximum height?b. What is the object’s maximum height?

Projectile Motion ProblemProjectile Motion ProblemAn object is launched An object is launched verticallyvertically upward from the ground at upward from the ground at

an initial velocity of 250 ft per second. an initial velocity of 250 ft per second.

10017516)( 2 tttsa.a. WhenWhen will the object reach its maximum height? will the object reach its maximum height?

(5.5, 578.5): Which is time? Which is height?(5.5, 578.5): Which is time? Which is height?b. What b. What is its maximum height? is its maximum height?

Projectile Motion ProblemProjectile Motion ProblemAn object is launched An object is launched verticallyvertically upward from the ground at upward from the ground at

an initial velocity of 250 ft per second. an initial velocity of 250 ft per second.

10017516)( 2 tttsc. Whenc. When will the object fall back to the ground? will the object fall back to the ground?

(11.5, 0) Which is time? Which is height?(11.5, 0) Which is time? Which is height?

Your turn:Your turn:An object is launched An object is launched verticallyvertically upward from the ground at upward from the ground at

an initial upward velocity of 123 ft per second. an initial upward velocity of 123 ft per second.

3. 3. When will the object fall back down to the ground?When will the object fall back down to the ground?

ttts 12316)( 2

1. 1. When will the object reach its maximum height?When will the object reach its maximum height?

2. 2. What is the object’s maximum height?What is the object’s maximum height?

Your turn:Your turn:4. 4. An object is launched An object is launched verticallyvertically dropped of the edge of a dropped of the edge of a

cliff. The cliff is 1500 feet high. When will the object fall cliff. The cliff is 1500 feet high. When will the object fall to the ground?to the ground?

150016)( 2 tts

5. 5. An airplane drops a bomb from an altitude of 20,000 An airplane drops a bomb from an altitude of 20,000 feet. How long does it take for the bomb to fall to the feet. How long does it take for the bomb to fall to the earth? earth?

000,2016)( 2 tts

Your turn:Your turn:6. 6. A snotty kid throws a water balloon off of his balcony at a A snotty kid throws a water balloon off of his balcony at a

girl below. The balloon’s initial vertical velocity is 10 feet girl below. The balloon’s initial vertical velocity is 10 feet per second downward. The balcony is 50 feet above the per second downward. The balcony is 50 feet above the sidewalk. When will the water balloon reach the sidewalk sidewalk. When will the water balloon reach the sidewalk beside the girl (he was a bad shot)? beside the girl (he was a bad shot)?

501016)( 2 ttts

Solve a totally non-recognizable quadratic Solve a totally non-recognizable quadratic equation.equation.

)2)(3(20 xx

)2)(3(0 xx By the zero product property, x = -3, 2By the zero product property, x = -3, 2

Can’t use the zero product propertyCan’t use the zero product property

620 2 xx

260 2 xx

Your turn:Your turn: 7. 7. Find the right side x-intercept for the Find the right side x-intercept for the following quadratic equation: following quadratic equation:

260 2 xx

Your turn:Your turn:8. 8. Write an expression for the following: “three more Write an expression for the following: “three more than twice a number”than twice a number”

9. 9. Write an expression for the following: “two less than Write an expression for the following: “two less than a number”a number”

Finding the dimensions of a Finding the dimensions of a rectangle.rectangle.

The length of one side of a rectangle is three more than The length of one side of a rectangle is three more than two times a number. What is the expression for the two times a number. What is the expression for the length of the side?length of the side?

x - 2x - 2

2x + 32x + 3

The width of the The width of the rectangle is two less rectangle is two less than the number. What than the number. What is the expression for the is the expression for the length of the side?length of the side?

If the area of the If the area of the rectangle is 400 square rectangle is 400 square inches, what is the inches, what is the length and width of the length and width of the rectangle?rectangle?

Finding the dimensions of a Finding the dimensions of a rectangle.rectangle.

x - 2x - 2

2x + 32x + 3

A = 400, length = ? Width = ?A = 400, length = ? Width = ?

WLA * 32 xL 2xwBy substitution:By substitution:

WL*400By substitution:By substitution:

Wx *)32(400 By substitution:By substitution:

)2)(32(400 xx

400A

Solve a totally non-recognizable quadratic Solve a totally non-recognizable quadratic equation.equation.

Can’t use the zero product propertyCan’t use the zero product property

Your turn:Your turn: 10. 10. Find the x-intercepts for the Find the x-intercepts for the following quadratic equation: following quadratic equation:

)2)(32(400 xx

)2)(32(400 xx

32 xL

2xw

11. 11. Use the values of x that you found in Use the values of x that you found in #10 to find the length of the rectangle.#10 to find the length of the rectangle.

12. 12. Use the values of x that you found in Use the values of x that you found in #10 to find the width of the rectangle.#10 to find the width of the rectangle.

VocabularyVocabularyx > 3 x > 3 andand x < 5 x < 5

HintHint: Inequality with “and” looks like: : Inequality with “and” looks like:

Compound inequalityCompound inequality

1 2 3 4 5 1 2 3 4 5

HintHint: This can also be written as: : This can also be written as: 3 < x < 53 < x < 5

Projectile motion:Projectile motion:A pebble is thrown upward with an intial velocity of 25 feet per A pebble is thrown upward with an intial velocity of 25 feet per second from ground level. second from ground level. ttts 2516)( 2

For what period of time willFor what period of time will it be it be above 5 feet?above 5 feet?

1t 2t

““and” type and” type inequalityinequality

Projectile motion:Projectile motion:A pebble is thrown upward with an intial velocity of 25 feet per A pebble is thrown upward with an intial velocity of 25 feet per second from ground level. second from ground level. ttts 2516)( 2

For what period of time willFor what period of time will it be it be above 5 feet?above 5 feet?

Projectile motion:Projectile motion:A pebble is thrown upward with an intial velocity of 25 feet per A pebble is thrown upward with an intial velocity of 25 feet per second from ground level. second from ground level. ttts 2516)( 2

For what period of time willFor what period of time will it be it be above 5 feet?above 5 feet?

1.33 sec1.33 sec0.24 sec0.24 sec

24.0 33.1

33.124.0 t

When above a certain When above a certain height?height?

tty 25016 2

An arrow is shot upward with an intial velocity of 250 feet per An arrow is shot upward with an intial velocity of 250 feet per second from ground level. For what period of time will the second from ground level. For what period of time will the arrow be above 1000 feet? arrow be above 1000 feet?

Never gets above Never gets above 1000 feet1000 feet

When above a certain When above a certain height?height?

tty 25016 2

An arrow is shot upward with an intial velocity of 250 feet per An arrow is shot upward with an intial velocity of 250 feet per second from ground level. For what period of time will the second from ground level. For what period of time will the arrow be above arrow be above 500 feet500 feet? ?

4.2 3.13

Your turn:Your turn:

A ball is dropped from the roof of a A ball is dropped from the roof of a 500 foot 500 foot building. building.

12. 12. When will it hit the ground? When will it hit the ground?

13. 13. For what period of time is the ball above 400 feet?For what period of time is the ball above 400 feet?

14. 14. When will the object be below 100 feet above ground When will the object be below 100 feet above ground level?level?

50016)( 2 tts

Summary:Summary:Projectile MotionProjectile Motion 1005016)( 2 ttts

1. Find the maximum height (1. Find the maximum height (y-value of the vertexy-value of the vertex))

2.2. Time to reach the maximum height (Time to reach the maximum height (x-value of the vertexx-value of the vertex))

3. Find the time to fall back to the ground (3. Find the time to fall back to the ground (x-interceptx-intercept))

Summary:Summary:4. Finding the dimensions of a box.4. Finding the dimensions of a box.

)32)(2(400 xxNon-standard quadratic equation.Non-standard quadratic equation.

Put into standard form and solve for ‘x’, then use ‘x’ to find the Put into standard form and solve for ‘x’, then use ‘x’ to find the length and the width of the box.length and the width of the box.

32

2

200

xlength

xwidth

Area

Summary:Summary:Projectile MotionProjectile Motion 1005016)( 2 ttts

5. Find the time period when the box is above 200 feet. 5. Find the time period when the box is above 200 feet.

Solve graphically.Solve graphically.

Use two equations. One of the equations is “y = 200”Use two equations. One of the equations is “y = 200”

Use Use intersectionsintersections to find when the object is exactly at 200 to find when the object is exactly at 200

Write the inequality for the value of time (x-values) when the Write the inequality for the value of time (x-values) when the object is above 200 feet.object is above 200 feet.