BLACK HISTORY MONTH

1
BLACK HISTORY MONTH

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

2
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 5x7y0
  represent New York Avenue and let 3x8y305
  represent Pennsylvania Avenue.
• Your mission today is to find the coordinates for
  the White House.

3
MISSION ACCOMPLISHED

• -5x7y0 3(5x7y)0
  -5x-175 -5x7(25)0
• 3x8y305 5(3x8y)305 x35
  -5x1750
• -15x21y0
• 15x40y1525
• 61y1525
• y 25
• The coordinate for the White House is (35,25).

4
FYI 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 200-meter dash, and the 4 100 meter
relay at the 1960 Olympic Games.

• Given
• 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 would run farther than the other
  runners. To compensate for this situation, the
  starting points of the runners are staggered.
• Your Mission
• 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.

5
Mission Accomplished

• Outside circle Inside circle
• Pi ( x 2.5) - Pi x 3.14x
  2.5(3.14) 3.14x
• 2.5
  (3.14)
• 7.85
• The two runners should start about 7.85 meters
  apart.

6
FYI 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 well as syndicated features in 145 newspapers.

• 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 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 spend
  21.50. Different theatures around town charge
  different amounts for adult and chid tickets.
• Your Mission
• Find out what price can the Johnsons and
  Oliveras afford to pay for each adult and each
  child .

7
Mission Accomplished

• Let a 1 adult ticket and
• c 1 child ticket
• 2a 4c 30 2a 4 (4.25)
  30
• -( 2a 2c 21.50 ) 2a 17 30
• 2c 8.5 - 17
  -17
• 2 2 2a
  13
• C 4.25 2
  2
• a
  6.50
• They can afford to pay 6.50 for an adult ticket
  and 4.25 for a childs ticket.

8
FYIThe Moscow Papyrus contains records of
early counting and measuring. It contains 25
problems. Of these, 11 concernproper portions
for making different strengths of beer and
bread. Given The longest loaf of bread ever
baked was 2,132 feet 2.5 inches. Todays
MissionIf this loaf were cut into .5 inch
slices, find out how many slices of bread would
there have been.
9
Mission Accomplished

• First change 2132 ft. 2.5 in. to inches.
• 2132 12 25,584
• 25,584 2.5 25,586.5 inches
• Next divide by .5
• 25,586 / .5 51,173
• There would have been 51,173 slices of bread.

10
FYIIn 1980, there were 1.2 million elephants
living in Africa. Because the natural 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 about 6.8 per year.

• Your Mission
• A) Write an equation to represent the population
  of the elephants in Africa.
• B) Find the year in which the population of
  Africa dropped by half of the population of the
  year 1980.

11
Mission Accomplished

• A) Y c(1-r)t
• 1.2(1-.086)t
• B) .6 gt 1.2( 1- .068)t
• 1.2 1.2
• .5 gt .932t
• Log .5 gt t log .932
• tgt log .5
• log .932
• tgt 9.8 year
• Towards the end of 1989 the elephant population
  dropped to less than half that of 1980.

12

• FYI The Babylonians computed ?2 to determine the
  diagonal of a square of side 1. This indicates,
  and other texts confirm that the Babylonians knew
  the Pythagorean theorem over a millennium before
  Pythagoras lived. They did not prove it. They
  knew that in a right or 90 triangle the sum of
  the squares of the lengths of the two shorter
  sides or legs (a²b²) equals the square of the
  length of the hypotenuse c².
• Your Mission
• Show your skill by picking up where the
  Babylonians left off.
• Prove the Pythagorean Theorem.

13
Mission Accomplished

• Given Right triangle ABC with right angle at C
• Prove a² b² c²
• Proof Draw the altitude from C to AB. Let AB
  c, ACb, BCa, ADx, DBy, and CDh.
• Two geometric means now exist c/a a/y and
  c/b b/x C
• 
  a²cy b² cx
  
• Add the equations
  b
  a
• a² b² cy cx
  
  h
• a² b² c(yx)
  
• a² b² c²
• 
  A
  B
• 
  x
  D y
• 
  c
  

14
FYI This was an actual problem from the Ahmose
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

15
Mission Accomplished

• 7777742401
• There were 2,401 going to Saint Ives.

16

• 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.

17
Mission Accomplished
18
The Berlin Papyrus

• FYI The Berlin Papyrus was created from
  fragments of early Egyptian tablets dating back
  prior to 1850 B.C. It solved simultaneous
  equations in verbal form.
• Your mission today is to solve 2 of the actual
  problems found on the Berlin Papyrus.
• x²y²100 x²y²400
• 4x-3y0 4x-3y0

19
Mission Accomplished

• X²y²100
  x²y²400
• 4x - 3y 0
  4x - 3y 0
• 
  x (3/4)y
  
• 4x 3y , x (3/4)y ,
• 4 4
• 
  
• (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
• ????y²/16) ????
  16 (25/16)y² 400
• (5/4)y 10
  ??2?y²) ?????
• 5y 40
  5y 80
• y8
  y 16
• X (3/4)(8)
  x (3/4)(16)
• X 6
  x 12

20

• FYI The Greek mathematician Thales was the first
  to measure the height of a pyramid by using
  geometry. He showed that the ratio of a pyramid
  to a staff was equal to the ratio of one shadow
  to the other.
• Given
• The height of a staff EF is 5 feet and the
  length of its shadow FD is 6 feet.
• Your mission today
• Find the height of a pyramid if the shadow of
  the pyramid BC is 576 feet, using similar
  triangles.

21
Mission Accomplishedx/576 5/6

6x 5(576) 6 6x 480 feet,
the height of the pyramid
22
FYI

• The Noks were one of Africas earliest
  civilizations, dating back to 500 B.C.
• Your mission today is to determine a valid
  conclusion can be reached from the two true
  statements. If it is a Nok sculpture, then it
  has hallowed out eyes and mouth and if a
  sculpture has hallowed out eyes and mouth, then
  it has air vents that prevent cracking.

23
Mission Accomplished

• Let p,q and r represent the parts of the
  statements.
• P it is a Nok sculpture
• Q it has hallowed out eyes and mouth
• R it has air vents that prevent cracking
• Using these letters, the given statements can be
  represented as p-q and q-r. Since the given
  statements are true, we can use the Law of
  Syllogism to conclude p-r. That is , if it is a
  Nok sculpture, then it has air vents that prevent
  cracking.

24
FYI Early Africans had technology for making
steel tools as early as 100 B.C.

• The gold disk once belonged to the high rulers
  of the West African kingdom of Ghana, which was
  founded about A.D. 200.
• If mlt140 ,mlt1 mlt2, and mlt2 mlt3, find m lt3.

25
Mission Accomplished

• Since congruence of angles is transitive, mlt1
  mlt3.
• Therefore, m lt340.

3
1
2
26
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

27
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 gt z,
  x z gt y, and y z gt x by the Triangle
  Inequality theorem.
• X Y X Triangle
• 1 1 10 no
• 1 2 9 no
• 1 3 8 no
• 1 4 7 no
• 1 5 6 no
• 2 2 8 no
• 2 3 7 no
• 2 4 6 no
• 2 5 5 yes
• 3 3 6 no
• 3 4 5 yes
• 4 4 4 yes
• There are 3 possible triangles.

28
FYI Problems 24 through 27 of the Ahmose
Papyrus give the oldest and most universal method
of solving linear equations with one unknown.

• Problem 26 states
• A quantity whose fourth part is
  added to it becomes 15.
• Your Mission
• Find the quantity.

29
Mission Accomplished

• x (x/4) 15
• 4xx/4 15
• 4xx 60
• 5x 60
• x12

Check 12 12(1/4) 12 3 15
30
FYI In 1942, the United States Army established
its first and only training facility for African
American pilots in Tuskegee, Alabama. In addition
to flying skills, pilots at the Tuskegee
Institute were taught to plot a course.

• Since distances are relative to the same
  airport the airport can be represented by the
  origin.
• An airplane at an elevation of 2 miles is 50
  miles east and 100 miles north of an airport.
  Another airplane at an elevation of 2.5 miles is
  240 miles west and 140 miles north of the
  airport.
• Your mission today is to find the distance
  between the two planes.

31
MISSION ACCOMPLISHED

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

The airplanes are about 292.75 miles apart.
32
FYI Benjamin Oliver Davis, Jr. was the first
African American to become an Air Force general
and command an air base.

• A plane P is 3 miles above an air base. The
  pilot sights the airport A at an angle of
  depression of 15 degrees. He sights his house H
  at an angle of depression of 32 degrees.
• Your Mission
• Find the ground distance d between the pilots
  house and the airport?

33
MISSION ACCOMPLISHED

• (sin 15)/5.66 (sin 17)/a
• a (sin 17 5.66)/sin 15
• a 6.4
• The distance between the pilots house and the
  airport is about 6.4 miles.

34
FYI Most of our knowledge of the order of
mathematics in Egypt is derived from two papyri-
The Rhind Papyrus and the Golenischev Papyrus.
The Rhind Papyrus was purchased in Luxor Egypt in
1858 by Scotman A. Henry Rhind.

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

35
MISSION ACCOMPLISHED

• Use the Pythagorean theorem to find the slant
  height
• l²250² (.5 646)²
• l²226,829
• l²v226,829476.3
• L½Pl
• ½(4646)(476.3)
• 615,379.6
• The area of the glass on the Luxor is about
  615,380ft²

36
FYI 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 performance,
Johnson used a computer model.

• The distance from his take off point to his
  landing point was 19.7 feet. The distance from
  the top of the hurdle to his landing point was
  8.8 feet.
• Your mission is to find the measure of the angle
  formed by his takeoff point to the top of the
  hurdle and the top of the hurdle to his landing
  point.

37
MISSION ACCOMPLISHED
38
FYI Bernard A. Harris Jr., MD is the 1st African
American astronaut to walk in space. In 1993, he
logged 4,164,183 miles in space on a mission
aboard the STS-55.

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

39
Mission Accomplished





Dr. Harris made about 156 orbits around
Earth.
40
FYI In the Old Babylonian period (1800-1600B.C.)
the circumference of a circle was found by taking
three times its diameter. Putting this equal to
pd, we see that their calculation is equivalent
to using 3 for the value of p.

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

41
MISSION ACCOMPLISHED

• Present day p Old Babylonian p
• 20,300 (2pr) 527,800(3)inches
• 20,300(2p)(13) 63,360 in/mi
• 527,800 p inches 24.99 miles
• For miles 12 5280 ft/mi or 63,360 in/mi
• 527,800p inches/63,360 inches/mi
• 26.17 miles

42
FYI 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 the
world. The area of the its base is 4050 square
meters.

• The volume of any pyramid is one-
• third the product of the area of the base B and
  its height h.

43
FYI continued

• Pyramid Height (
  meters)
• Great Pyramid of Cheops 147
• Bent Pyramid in Egypt 101
• Inca Pyramid in Peru 75
• Pyramid of the Sun in Mexico 60
• Step Pyramid of Djoser in Egypt 60
• Pyramid of the Sun in Peru 50
• Write an expression that represents the volume of
  a pyramid.
• b) Find the volume of the Great Pyramid of Cheops

44
MISSION ACCMPLISHED

• a) V1/3 Bh
• b) V4050(147)
• 3
• V198,450 cu. ft.

45
FYI In 1995, Sheila Hudson Strudwick set the
U.S. womens triple jump record.

• Suppose her first jump was 46 feet 9 inches and
  her next jump was 1 ¾ inches longer. Her
  record-setting jump was 1 foot 2.5 inches longer
  than her second jump.
• Your mission Find the length of her record
  setting jump.

46
MISSION ACCOMPLISHED

• 1st jump 46ft 9in
• 2nd jump 46ft 9in 1 ¾ 46ft 10 ¾in
• 3rd jump 46ft 10 ¾in 1ft ½ in
• 46ft 10 ¾ in
• 1ft 2 ½ in
• 47 ft 12 5/4in
• Her record setting jump was
  48ft 1 ¼in

47
FYI The largest loaf of bread ever baked was
2132 feet 2 ½in.

• If this loaf were cut into ½ slices, your
  mission is to find out how many slices of bread
  would there have been?

48
MISSION ACCOMPLISHED

• 1st change 2132ft 2 ½ to inches
• 213212 25584
• 2 ½
• 25,586.5
• Then divide by ½
• 25,586.5/.551,173
• There would have been 51,173 slices of bread.

49
FYI Truncated square pyramids, which are
shaped like cornerstones with sloping sides
required great accuracy in constructing the early
pyramids. Egyptian scribal mathematicians had
found at least one specific case of the correct
volume formula V(h/3)
(a2abb2)

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

50
Mission Accomplished

• V(h/3)(a2abb2)
• V(6/3)(222(4)42)
• V2(4816)
• V2(28)
• V56