Download ppt - BLACK HISTORY MONTH

BLACK HISTORY MONTHBLACK HISTORY MONTH

These problems were created to be used on the These problems were created to be used on the High School level as warm-up exercises or to fill High School level as warm-up exercises or to fill in time at the end of a class. There are enough in time at the end of a class. There are enough exercise to take you through 4 weeks with an exercise to take you through 4 weeks with an extra 3 problems included. They range from extra 3 problems included. They range from algebra to trigonometry. I hope this will not only algebra to trigonometry. I hope this will not only be educational, but fun!be educational, but fun!

Enjoy your mission…Enjoy your mission… Ms. Valerie Russell Ms. Valerie Russell

FYI: Geography: Benjamin Banneker, a self taught

mathematician, was the first African-American to publish an almanac. He is most noted for being the assistant surveyor on the team that

designed the ten-mile square of Washington D.C.

The White House is located in the center of the square, at the intersection of Pennsylvania Avenue and New York Avenue. Let –5x+7y=0 represent New York Avenue and let 3x+8y=305 represent Pennsylvania Avenue.

Your mission today is to find the coordinates for the White House.

MISSION ACCOMPLISHED-5x+7y=0 3(5x+7y)=0 -5x=-175 -5x+7(25)=0 3x+8y=305 5(3x+8y)=305 x=35 -5x+175=0

-15x+21y=0 15x+40y=1525 61y=1525 y = 25

The coordinate for the White House is (35,25).

FYI: FYI: Wilma Rudolf (1940-1994) was the first African-American woman to win three Wilma Rudolf (1940-1994) was the first African-American woman to win three gold medals at a single Olympic games. She won the 100-meter sprint, the gold medals at a single Olympic games. She won the 100-meter sprint, the 200-meter dash, and the 4* 100 meter relay at the 1960 Olympic Games.200-meter dash, and the 4* 100 meter relay at the 1960 Olympic Games.

Given:Given: The runners in a 200 meter dash race ran around the curve of a track. If The runners in a 200 meter dash race ran around the curve of a track. If

the runners start and finish at the same line, the runner on the outside lane the runners start and finish at the same line, the runner on the outside lane would run farther than the other runners. To compensate for this situation, would run farther than the other runners. To compensate for this situation, the starting points of the runners are staggered. the starting points of the runners are staggered.

Your Mission:Your Mission:

If the radius of the inside lane is X and each lane is 2.5 ft. wide, find out how If the radius of the inside lane is X and each lane is 2.5 ft. wide, find out how far apart should the officials start the runners in the two inside lanes.far apart should the officials start the runners in the two inside lanes.

Mission AccomplishedMission Accomplished

Outside circle Inside circleOutside circle Inside circle PiPi ( x+ 2.5) - ( x+ 2.5) - Pi Pi x = 3.14x + 2.5(3.14) – 3.14x x = 3.14x + 2.5(3.14) – 3.14x = 2.5 (3.14)= 2.5 (3.14) = 7.85= 7.85

The two runners should start about 7.85 meters apart.The two runners should start about 7.85 meters apart.

FYI: FYI: Elmer Simms Campbell was the first African-American cartoonist to work for national Elmer Simms Campbell was the first African-American cartoonist to work for national

publications. He contributed cartoons and other art work to Esquire, Cosmopolitan, and Redbook, as publications. He contributed cartoons and other art work to Esquire, Cosmopolitan, and Redbook, as well as syndicated features in 145 newspapers.well as syndicated features in 145 newspapers.

Given:Given: The Lion King was the top-grossing film of the year, making an estimated 300.4 million at the box office. On a Saturday The Lion King was the top-grossing film of the year, making an estimated 300.4 million at the box office. On a Saturday

afternoon the Johnson and Olivera families decided to go see The Lion King together. The Johnson family, two adults and afternoon the Johnson and Olivera families decided to go see The Lion King together. The Johnson family, two adults and four children can afford to spend $30 on movie tickets while the Olivera family, two adults and two children can afford to four children can afford to spend $30 on movie tickets while the Olivera family, two adults and two children can afford to spend $21.50. Different theatures around town charge different amounts for adult and chid tickets.spend $21.50. Different theatures around town charge different amounts for adult and chid tickets.

Your Mission:Your Mission: Find out what price can the Johnson’s and Olivera’s afford to pay for each adult and each childFind out what price can the Johnson’s and Olivera’s afford to pay for each adult and each child . .


Let a = 1 adult ticket andLet a = 1 adult ticket and c = 1 child ticketc = 1 child ticket

2a + 4c = 30 2a +4 (4.25) = 302a + 4c = 30 2a +4 (4.25) = 30 -( 2a + 2c = 21.50-( 2a + 2c = 21.50 ) 2a + 17 = 30 ) 2a + 17 = 30 2c 2c = = 8.58.5 - 17 -17- 17 -17 2 2 2 2 2a 2a = = 13 13 C= $4.25 2 2C= $4.25 2 2 a = $6.50 a = $6.50 They can afford to pay $6.50 for an adult ticket and $4.25 for a They can afford to pay $6.50 for an adult ticket and $4.25 for a

childs ticket.childs ticket.

FYI:FYI:The Moscow Papyrus contains records of early counting and The Moscow Papyrus contains records of early counting and

measuring. It contains 25 problems. Of these, 11 concernmeasuring. It contains 25 problems. Of these, 11 concernproper portions for making different strengths of beer and bread.proper portions for making different strengths of beer and bread.

Given: The longest loaf of bread ever baked was 2,132 feet 2.5 Given: The longest loaf of bread ever baked was 2,132 feet 2.5

inches. inches.

Today’s Mission:Today’s Mission:If this loaf were cut into .5 inch slices, find out how many slices If this loaf were cut into .5 inch slices, find out how many slices

of bread would there have been.of bread would there have been.


First change 2132 ft. 2.5 in. to inches.First change 2132 ft. 2.5 in. to inches. 2132 * 12 = 25,5842132 * 12 = 25,584 25,584 + 2.5 = 25,586.5 inches25,584 + 2.5 = 25,586.5 inchesNext divide by .5Next divide by .525,586 / .5 = 51,17325,586 / .5 = 51,173There would have been 51,173 slices of There would have been 51,173 slices of

bread.bread.

FYI:FYI:In 1980, there were 1.2 million elephants living in Africa. Because the natural In 1980, there were 1.2 million elephants living in Africa. Because the natural grazing lands for the elephant are disappearing due to increased population grazing lands for the elephant are disappearing due to increased population

and cultivation of the land, the number of elephants in Africa has decreased by and cultivation of the land, the number of elephants in Africa has decreased by about 6.8% per year.about 6.8% per year.

Your Mission:Your Mission: A) Write an equation to A) Write an equation to

represent the population represent the population of the elephants in of the elephants in Africa.Africa.

B) Find the year in B) Find the year in which the population of which the population of Africa dropped by half of Africa dropped by half of the population of the the population of the year 1980. year 1980.

Mission AccomplishedMission Accomplished A) Y= c(1-r)^tA) Y= c(1-r)^t = 1.2(1-.086)^t= 1.2(1-.086)^t

B) B) .6.6 > > 1.21.2( 1- .068)^t ( 1- .068)^t 1.2 1.2 1.2 1.2 .5 > .932^t.5 > .932^t Log .5 > t log .932Log .5 > t log .932 t> t> log .5log .5 log .932log .932 t> 9.8 yeart> 9.8 year Towards the end of 1989 the elephant population dropped to less Towards the end of 1989 the elephant population dropped to less

than half that of 1980.than half that of 1980.

€

1.2(1 − .086)

€

t

€

1.2 1− .086( )t

€ € €

1.2 1− .086( )t

€

1.2 1− .086( )t

FYIFYI: : The Babylonians computed The Babylonians computed 2 to determine the 2 to determine the

diagonal of a square of side 1. This indicates, and other diagonal of a square of side 1. This indicates, and other texts confirm that the Babylonians knew the Pythagorean texts confirm that the Babylonians knew the Pythagorean theorem over a millennium before Pythagoras lived. They theorem over a millennium before Pythagoras lived. They did not prove it. They knew that in a right or 90did not prove it. They knew that in a right or 90° triangle ° triangle the sum of the squares of the lengths of the two shorter the sum of the squares of the lengths of the two shorter sides or legs (a²+b²) equals the square of the length of the sides or legs (a²+b²) equals the square of the length of the hypotenuse c².hypotenuse c².

Your Mission:Your Mission:Show your skill by picking up where the Babylonians left off.Show your skill by picking up where the Babylonians left off.

Prove the Pythagorean Theorem.Prove the Pythagorean Theorem.

Mission AccomplishedMission AccomplishedGiven: Right triangle ABC with right angle at CGiven: Right triangle ABC with right angle at C

Prove: a² + b² = c²Prove: a² + b² = c²

Proof: Draw the altitude from C to AB. Let AB = c, AC=b, BC=a, AD=x, DB=y, and CD=h.Proof: Draw the altitude from C to AB. Let AB = c, AC=b, BC=a, AD=x, DB=y, and CD=h.

Two geometric means now exist: c/a = a/y and c/b = b/x CTwo geometric means now exist: c/a = a/y and c/b = b/x C

a²=cy b²= cx a²=cy b²= cx

Add the equations: b aAdd the equations: b a

a² + b² = cy + cx h a² + b² = cy + cx h

a² + b² = c(y+x) a² + b² = c(y+x)

a² + b² = c²a² + b² = c²

A BA B

x D y x D y

c c

FYI: This was an actual problem from the A’hmose Papyrus.

It is considered the antecedent to the famous Mother Goose rhyme .

As I was going to Saint Ives.I met a man with 7 wives. Each wife had 7 sacks;Every sack had seven cats;Every cat had seven kits;Kits, cats, sacks, and wives,How many were going to Saint Ives

Mission Accomplished

7*7*7*7=7^4=2401There were 2,401 going to Saint

Ives.

FYI: In 2000 BC, Egyptians used a loop of rope knotted at 12 equal

intervals to determine the boundaries of their properties each year after the Nile flooded! They stretched the rope around three steaks so that the measures of the sides of the triangle we 3,4, and 5. This right triangle was used to establish the corners of their fields.

The 3,4,5 right triangle is called a Pythagorean triple. Pretend you lived back during the days of the Egyptians and you owned 4 pieces of property, all of which had to be bounded by right triangles.

Your Mission:

Find 4 other triples that you can use to tie off your property.

Mission Accomplished

The Berlin PapyrusThe Berlin Papyrus

FYIFYI:: The Berlin Papyrus was created from The Berlin Papyrus was created from fragments of early Egyptian tablets dating back fragments of early Egyptian tablets dating back prior to 1850 B.C. It solved simultaneous prior to 1850 B.C. It solved simultaneous equations in verbal form.equations in verbal form.

**Your mission today is to solve 2 of the Your mission today is to solve 2 of the actual problems found on the Berlin actual problems found on the Berlin Papyrus.Papyrus.

xx²+y²=100 ²+y²=100 xx²+y²=400²+y²=400 4x-3y=0 4x-3y=04x-3y=0 4x-3y=0

Mission AccomplishedMission AccomplishedXX²+y²=100 x²+y²=400²+y²=100 x²+y²=400

4x - 3y = 0 4x - 3y = 04x - 3y = 0 4x - 3y = 0

x = (3/4)y x = (3/4)y

4x4x = = 3y3y , x = (3/4)y , , x = (3/4)y ,

4 44 4

(3/4 y)² + y² =100 (3/4 y)² + y² = 400(3/4 y)² + y² =100 (3/4 y)² + y² = 400

(9/16)y² + (16/16)y² = 100 (9/16)y + 16/16)y² = 400(9/16)y² + (16/16)y² = 100 (9/16)y + 16/16)y² = 400

y²/16) = y²/16) = 16 [ (25/16)y² = 400 ] 16 [ (25/16)y² = 400 ]

(5/4)y = 10 (5/4)y = 10 22y²) =y²) =5y = 40 5y = 805y = 40 5y = 80

y=8 y = 16y=8 y = 16

X = (3/4)(8) x = (3/4)(16)X = (3/4)(8) x = (3/4)(16)

X = 6 x = 12 X = 6 x = 12

FYI:FYI: The Greek mathematician The Greek mathematician Thales was the first to measure the Thales was the first to measure the height of a pyramid by using geometry. height of a pyramid by using geometry. He showed that the ratio of a pyramid He showed that the ratio of a pyramid to a staff was equal to the ratio of one to a staff was equal to the ratio of one shadow to the other.shadow to the other.

Given:Given: The height of a staff EF is 5 feet The height of a staff EF is 5 feet

and the length of its shadow FD is and the length of its shadow FD is 6 feet.6 feet.

Your mission today: Your mission today: Find the height of a pyramid if the Find the height of a pyramid if the

shadow of the pyramid BC is 576 shadow of the pyramid BC is 576 feet, using similar triangles.feet, using similar triangles.


x/576 = 5/6x/576 = 5/6

6x 6x = = 5(576) 5(576) 6 6 6 6

x = 480 feet, the height of the pyramid x = 480 feet, the height of the pyramid

FYI:FYI:

The Noks were one of Africa’s earliest The Noks were one of Africa’s earliest civilizations, dating back to 500 B.C.civilizations, dating back to 500 B.C.

Your mission today is to determine a Your mission today is to determine a valid conclusion can be reached valid conclusion can be reached from the two true statements. If it is from the two true statements. If it is a Nok sculpture, then it has hallowed a Nok sculpture, then it has hallowed out eyes and mouth and if a out eyes and mouth and if a sculpture has hallowed out eyes and sculpture has hallowed out eyes and mouth, then it has air vents that mouth, then it has air vents that prevent cracking.prevent cracking.


Let p,q and r represent the parts of the Let p,q and r represent the parts of the statements.statements.

P: it is a Nok sculptureP: it is a Nok sculptureQ: it has hallowed out eyes and mouthQ: it has hallowed out eyes and mouthR: it has air vents that prevent crackingR: it has air vents that prevent cracking*Using these letters, the given statements *Using these letters, the given statements

can be represented as p-q and q-r. Since can be represented as p-q and q-r. Since the given statements are true, we can the given statements are true, we can use the Law of Syllogism to conclude p-r. use the Law of Syllogism to conclude p-r. That is , if it is a Nok sculpture, then it That is , if it is a Nok sculpture, then it has air vents that prevent cracking.has air vents that prevent cracking.

FYIFYI:: Early Africans had technology for making steel Early Africans had technology for making steel

tools as early as 100 B.C.tools as early as 100 B.C.

The gold disk once belonged to the The gold disk once belonged to the high rulers of the West African high rulers of the West African kingdom of Ghana, which was kingdom of Ghana, which was founded about A.D. 200.founded about A.D. 200.

If mIf m<1=40 ,m<1= m<2, and m<2 = <1=40 ,m<1= m<2, and m<2 = m<3, find m <3.m<3, find m <3.


Since congruence of Since congruence of angles is transitive, angles is transitive, mm<1= m<3.<1= m<3.

Therefore, m <3=40.Therefore, m <3=40.

12

3

FYI: The early Egyptians used to make triangles by using a rope with knots tied at equal intervals. Each vertex of the triangle had to occur at a knot. Suppose you had a rope with exactly 13

knots.

Your Mission:Find out how many different triangles you can

make

MISSION ACCOMPLISHED

The 13 knots divide the rope into 12 sections of equal length. Thus, the perimeter of each rectangle that can be formed will be 12 units. If x, y, and z represent the lengths of the 3 sides then x + y + z= 12. We also know that x + y > z, x + z > y, and y + z > x by the Triangle Inequality theorem.

X Y X Triangle1 1 10 no1 2 9 no1 3 8 no1 4 7 no1 5 6 no2 2 8 no2 3 7 no2 4 6 no2 5 5 yes3 3 6 no3 4 5 yes4 4 4 yes

There are 3 possible triangles.

FYI: FYI: Problems 24 through 27 of the A’hmose Papyrus Problems 24 through 27 of the A’hmose Papyrus

give the oldest and most universal method of solving linear give the oldest and most universal method of solving linear equations with one unknown.equations with one unknown.

Problem 26 states:Problem 26 states: “ “A quantity whose fourth part is A quantity whose fourth part is

added to it becomes 15.” added to it becomes 15.”

Your Mission:Your Mission:Find the quantity.Find the quantity.


x + (x/4) = 15x + (x/4) = 15

4[x+x/4 = 15]4[x+x/4 = 15]

4x+x = 604x+x = 60

5x = 605x = 60

x=12x=12

Check: 12 + 12(1/4) = 12 + 3 = 15

FYI: FYI: In 1942, the United States Army established its first In 1942, the United States Army established its first

and only training facility for African American pilots in and only training facility for African American pilots in Tuskegee, Alabama. In addition to flying skills, pilots at the Tuskegee, Alabama. In addition to flying skills, pilots at the

Tuskegee Institute were taught to plot a course.Tuskegee Institute were taught to plot a course. Since distances are relative to the same airport the Since distances are relative to the same airport the

airport can be represented by the origin.airport can be represented by the origin.

An airplane at an elevation of 2 miles is 50 miles An airplane at an elevation of 2 miles is 50 miles east and 100 miles north of an airport. Another east and 100 miles north of an airport. Another airplane at an elevation of 2.5 miles is 240 miles airplane at an elevation of 2.5 miles is 240 miles west and 140 miles north of the airport. west and 140 miles north of the airport.

Your mission today is to find the distance between Your mission today is to find the distance between the two planes.the two planes.

MISSION ACCOMPLISHEDMISSION ACCOMPLISHED

Use the distance between the airplanes given 3 points in Use the distance between the airplanes given 3 points in space (x, y, z). The distance between A and B isspace (x, y, z). The distance between A and B is

75.292

25.700,85

5.040290

5.2214010024050

222

222

212

212

2

12

xxyyxxAB

The airplanes are about 292.75 miles apart.

FYI: FYI: Benjamin Oliver Davis, Jr. was the first African Benjamin Oliver Davis, Jr. was the first African

American to become an Air Force general and command American to become an Air Force general and command an air base.an air base.

A plane P is 3 miles above an air base. A plane P is 3 miles above an air base. The pilot sights the airport A at an The pilot sights the airport A at an angle of depression of 15 degrees. angle of depression of 15 degrees. He sights his house H at an angle of He sights his house H at an angle of depression of 32 degrees. depression of 32 degrees.


Find the ground distance d between Find the ground distance d between the pilots house and the airport?the pilots house and the airport?


(sin 15°)/5.66 = (sin 17°)/a(sin 15°)/5.66 = (sin 17°)/a a = (sin 17° * 5.66)/sin 15°a = (sin 17° * 5.66)/sin 15° a = 6.4a = 6.4

The distance between the pilots The distance between the pilots house and the airport is about 6.4 house and the airport is about 6.4 miles.miles.

FYI: FYI: Most of our knowledge of the order of Most of our knowledge of the order of

mathematics in Egypt is derived from two papyri- The mathematics in Egypt is derived from two papyri- The Rhind Papyrus and the Golenischev Papyrus. The Rhind Rhind Papyrus and the Golenischev Papyrus. The Rhind Papyrus was purchased in Luxor Egypt in 1858 by Papyrus was purchased in Luxor Egypt in 1858 by Scotman A. Henry Rhind.Scotman A. Henry Rhind.

The Luxor Hotel in Las Vegas, Nevada, is The Luxor Hotel in Las Vegas, Nevada, is shaped like a gigantic black glass pyramid. shaped like a gigantic black glass pyramid. The base of the pyramid is a square with The base of the pyramid is a square with edges 646 feet long. The hotel is 350 feet tall. edges 646 feet long. The hotel is 350 feet tall.

Your mission:Your mission: Find the area of the glass on the Luxor.Find the area of the glass on the Luxor.*The area of the glass is the lateral area of the *The area of the glass is the lateral area of the

pyramid.pyramid.


Use the Pythagorean theorem to find the slant height:Use the Pythagorean theorem to find the slant height: ll²=250² + (.5 * 646)²²=250² + (.5 * 646)² l²=226,829l²=226,829 l²=√226,829≈476.3l²=√226,829≈476.3

L=½PlL=½Pl =½(4*646)(476.3)=½(4*646)(476.3) ≈ ≈615,379.6615,379.6 The area of the glass on the Luxor is about 615,380ft² The area of the glass on the Luxor is about 615,380ft²

FYIFYI: : In 1996, Allen Johnson ran the 110-meter hurdles in 12.92 seconds making him the In 1996, Allen Johnson ran the 110-meter hurdles in 12.92 seconds making him the fourth athlete to run this event in less than 13 seconds. To help him improve his fourth athlete to run this event in less than 13 seconds. To help him improve his

performance, Johnson used a computer model.performance, Johnson used a computer model.

The distance from his take off point to his The distance from his take off point to his landing point was 19.7 feet. The distance landing point was 19.7 feet. The distance from the top of the hurdle to his landing from the top of the hurdle to his landing point was 8.8 feet.point was 8.8 feet.

Your mission is to find the measure of the Your mission is to find the measure of the angle formed by his takeoff point to the angle formed by his takeoff point to the top of the hurdle and the top of the hurdle top of the hurdle and the top of the hurdle to his landing point.to his landing point.


€

t 2 = s2 + t 2 − 2st cosT

19.72 = 8.82 +12.12 − 2 8.8( ) 12.1( )cosT

19.72 − 8.82 −12.12

−2 8.8( ) 12.1( )= cosT

−0.7712 ≈ cosT

140.5 ≈ T

FYI: FYI: Bernard A. Harris Jr., MD is the 1Bernard A. Harris Jr., MD is the 1stst African African American astronaut to walk in space. In 1993, he American astronaut to walk in space. In 1993, he

logged 4,164,183 miles in space on a mission logged 4,164,183 miles in space on a mission

aboard the STS-55.aboard the STS-55.

If the orbit was 250 miles above If the orbit was 250 miles above Earth and the diameter of Earth is Earth and the diameter of Earth is about 8000 miles, your mission today about 8000 miles, your mission today is to find how many orbits around is to find how many orbits around Earth Dr. Harris made on that Earth Dr. Harris made on that mission.mission.

Mission Accomplished Mission Accomplished

Dr. Harris made about 156 orbits around Earth.€

C = πd, d = 8000 + 500

4164,183

8500π=

8500πx

8500πx =155.94

FYI: FYI: In the Old Babylonian period (1800-1600B.C.) the In the Old Babylonian period (1800-1600B.C.) the

circumference of a circle was found by taking three times its circumference of a circle was found by taking three times its diameter. Putting this equal to diameter. Putting this equal to πd, we see that their πd, we see that their calculation is equivalent to using 3 for the value of π.calculation is equivalent to using 3 for the value of π.

In 1996, inspectors checked the accuracy of the In 1996, inspectors checked the accuracy of the Olympic marathon course by using the Olympic marathon course by using the “calibrated bicycle method.” The inspectors “calibrated bicycle method.” The inspectors rode bicycles equipped with counters that rode bicycles equipped with counters that tracked the turns of the wheels. If the radius tracked the turns of the wheels. If the radius of a bicycle wheel is 13 inches and the of a bicycle wheel is 13 inches and the counter shows 20,300 revolutions at the end counter shows 20,300 revolutions at the end of a course, your mission is to find the length of a course, your mission is to find the length of the course using the Old Babylonian value of the course using the Old Babylonian value of piof pi and the present day value of pi. and the present day value of pi.


Present day Present day π Old Babylonian ππ Old Babylonian π

20,300 (2πr) 20,300 (2πr) 527,800(3)inches527,800(3)inches

=20,300(2π)(13) 63,360 in/mi=20,300(2π)(13) 63,360 in/mi

=527,800 π inches =24.99 miles=527,800 π inches =24.99 miles

For miles 12” = 5280 ft/mi or 63,360 in/miFor miles 12” = 5280 ft/mi or 63,360 in/mi

527,800π inches/63,360 inches/mi527,800π inches/63,360 inches/mi

=26.17 miles=26.17 miles

FYI: FYI: The Great Pyramid of Cheops in Egypt is considered to be The Great Pyramid of Cheops in Egypt is considered to be

one of the Seven Wonders of the World. It is also the largest pyramid in one of the Seven Wonders of the World. It is also the largest pyramid in

the world. The area of the its base is 4050 square meters.the world. The area of the its base is 4050 square meters.

The volume of any pyramid is one-The volume of any pyramid is one-

third the product of the area of the third the product of the area of the base B and its height h.base B and its height h.

FYI continuedFYI continued

PyramidPyramid Height ( metersHeight ( meters))Great Pyramid of CheopsGreat Pyramid of Cheops 147 147Bent Pyramid in EgyptBent Pyramid in Egypt 101 101Inca Pyramid in PeruInca Pyramid in Peru 75 75Pyramid of the Sun in Mexico 60Pyramid of the Sun in Mexico 60Step Pyramid of Djoser in Egypt 60Step Pyramid of Djoser in Egypt 60Pyramid of the Sun in Peru 50Pyramid of the Sun in Peru 50a)a) Write an expression that represents the Write an expression that represents the

volume of a pyramid.volume of a pyramid.bb) Find the volume of the Great Pyramid of ) Find the volume of the Great Pyramid of

CheopsCheops

MISSION ACCMPLISHEDMISSION ACCMPLISHED

a) V=1/3 Bha) V=1/3 Bh

b) V=b) V=4050(147)4050(147)

33

V=198,450 cu. ft.V=198,450 cu. ft.

FYI: FYI: In 1995, Sheila Hudson Strudwick set In 1995, Sheila Hudson Strudwick set

the U.S. women’s triple jump record.the U.S. women’s triple jump record. Suppose her first jump was 46 feet 9 inches Suppose her first jump was 46 feet 9 inches

and her next jump was 1 ¾ inches longer. and her next jump was 1 ¾ inches longer. Her record-setting jump was 1 foot 2.5 Her record-setting jump was 1 foot 2.5 inches longer than her second jump. inches longer than her second jump.

Your mission: Find the length of her record Your mission: Find the length of her record setting jump.setting jump.

MISSION MISSION ACCOMPLISHEDACCOMPLISHED

11stst jump – 46ft 9in jump – 46ft 9in

22ndnd jump – 46ft 9in + 1 ¾ = 46ft 10 ¾in jump – 46ft 9in + 1 ¾ = 46ft 10 ¾in

33rdrd jump – 46ft 10 ¾in + 1ft ½ in = jump – 46ft 10 ¾in + 1ft ½ in =

46ft 10 ¾ in46ft 10 ¾ in

+ + 1ft 2 ½ in1ft 2 ½ in

47 ft 12 5/4in47 ft 12 5/4in

Her record setting jump was Her record setting jump was 48ft 1 ¼in 48ft 1 ¼in

FYI: FYI: The largest loaf of bread ever baked was 2132 The largest loaf of bread ever baked was 2132

feet 2 ½in.feet 2 ½in. If this loaf were cut into ½” slices, If this loaf were cut into ½” slices,

your mission is to find out how many your mission is to find out how many slices of bread would there have slices of bread would there have been?been?


11stst change 2132ft 2 ½” to inches change 2132ft 2 ½” to inches

2132*12 = 255842132*12 = 25584

+ + 2 ½”2 ½”

25,586.5”25,586.5”

Then divide by ½:Then divide by ½:

25,586.5/.5=51,17325,586.5/.5=51,173

*There would have been 51,173 slices of *There would have been 51,173 slices of bread.bread.

FYI: FYI: TruncatedTruncated square pyramids, which are shaped like square pyramids, which are shaped like

cornerstones with sloping sides required great accuracy in cornerstones with sloping sides required great accuracy in constructing the early pyramids. Egyptian scribal constructing the early pyramids. Egyptian scribal

mathematicians had found at least one specific case of the mathematicians had found at least one specific case of the correct volume formula :correct volume formula :

V=(h/3) (a^2+ab+b^2) V=(h/3) (a^2+ab+b^2)


Find the volume of a truncated square Find the volume of a truncated square pyramid (or frustum) with altitude 6, pyramid (or frustum) with altitude 6, base side 4, and top side 2.base side 4, and top side 2.


V=(h/3)(a^2+ab+b^2)V=(h/3)(a^2+ab+b^2)

V=(6/3)(2^2+2(4)+4^2)V=(6/3)(2^2+2(4)+4^2)

V=2(4+8+16)V=2(4+8+16)

V=2(28)V=2(28)

V=56V=56