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

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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.

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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).

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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: 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. 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: 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. 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.

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Mission Accomplished Outside circle Inside circle Outside circle Inside circle Pi ( x+ 2.5) - Pi x = 3.14x + 2.5(3.14) – 3.14x Pi ( x+ 2.5) - Pi 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.

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

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

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FYI: The Moscow Papyrus contains records of early counting and measuring. It contains 25 problems. Of these, 11 concern proper portions for making different strengths of beer and bread. Given: The longest loaf of bread ever baked was 2,132 feet 2.5 inches. Today’s Mission: If this loaf were cut into.5 inch slices, find out how many slices of bread would there have been.

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

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FYI: 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 and cultivation of the land, the number of elephants in Africa has decreased by about 6.8% per year. Your Mission: Your Mission: A) Write an equation to represent the population of the elephants in Africa. 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. B) Find the year in which the population of Africa dropped by half of the population of the year 1980.

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Mission Accomplished A) Y= c(1-r)^t A) Y= c(1-r)^t = 1.2(1-.086)^t = 1.2(1-.086)^t B).6 > 1.2( 1-.068)^t B).6 > 1.2( 1-.068)^t 1.2 1.2 1.2 1.2.5 >.932^t.5 >.932^t Log.5 > t log.932 Log.5 > t log.932 t> log.5 t> log.5 log.932 log.932 t> 9.8 year t> 9.8 year Towards the end of 1989 the elephant population dropped to less than half that of 1980. Towards the end of 1989 the elephant population dropped to less than half that of 1980.

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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 (a²+b²) 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. Prove the Pythagorean Theorem.

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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, 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 C a²=cy b²= cx a²=cy b²= cx Add the equations: b a a² + b² = cy + cx h a² + b² = c(y+x) a² + b² = c² A B A B x D y x D y c

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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

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Mission Accomplished 7*7*7*7=7^4=2401 There were 2,401 going to Saint Ives.

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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.

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Mission Accomplished

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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. x²+y²=100 x²+y²=400 x²+y²=100 x²+y²=400 4x-3y=0 4x-3y=0 4x-3y=0 4x-3y=0

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Mission Accomplished X²+y²=100 x²+y²=400 4x - 3y = 0 4x - 3y = 0 x = (3/4)y 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 2y²) = 5y = 40 5y = 80 y=8 y = 16 X = (3/4)(8) x = (3/4)(16) X = 6 x = 12

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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. 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. Find the height of a pyramid if the shadow of the pyramid BC is 576 feet, using similar triangles.

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Mission Accomplished x/576 = 5/6 6x = 5(576) 6 6 x = 480 feet, the height of the pyramid Mission Accomplished x/576 = 5/6 6x = 5(576) 6 6 x = 480 feet, the height of the pyramid

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FYI: The Noks were one of Africa’s 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.

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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.

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FYI : Early Africans had technology for making steel tools as early as 100 B.C. 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. The gold disk once belonged to the high rulers of the West African kingdom of Ghana, which was founded about A.D. 200. If m<1=40,m<1= m<2, and m<2 = m<3, find m <3.

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Mission Accomplished Mission Accomplished Since congruence of angles is transitive, m<1= m<3. Therefore, m <3=40. 1 2 3

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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

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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 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.

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FYI: Problems 24 through 27 of the A’hmose 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.” “A quantity whose fourth part is added to it becomes 15.” Your Mission: Find the quantity.

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Mission Accomplished x + (x/4) = 15 x + (x/4) = 15 4[x+x/4 = 15] 4[x+x/4 = 15] 4x+x = 60 4x+x = 60 5x = 60 5x = 60 x=12 x=12 Check: 12 + 12(1/4) = 12 + 3 = 15

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

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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.

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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?

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MISSION ACCOMPLISHED 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.

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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. 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. Find the area of the glass on the Luxor. *The area of the glass is the lateral area of the pyramid.

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MISSION ACCOMPLISHED MISSION ACCOMPLISHED Use the Pythagorean theorem to find the slant height: l²=250² + (.5 * 646)² l²=226,829 l²=√226,829≈476.3 L=½Pl =½(4*646)(476.3) ≈615,379.6 The area of the glass on the Luxor is about 615,380ft²

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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.

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MISSION ACCOMPLISHED MISSION ACCOMPLISHED

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FYI: Bernard A. Harris Jr., MD is the 1 st 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. 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.

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Mission Accomplished Mission Accomplished Dr. Harris made about 156 orbits around Earth.

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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 πd, we see that their calculation is equivalent to using 3 for the value of π. 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.

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MISSION ACCOMPLISHED MISSION ACCOMPLISHED Present day π Old Babylonian π 20,300 (2πr) 527,800(3)inches =20,300(2π)(13) 63,360 in/mi =527,800 π inches =24.99 miles For miles 12” = 5280 ft/mi or 63,360 in/mi 527,800π inches/63,360 inches/mi =26.17 miles

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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.

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FYI continued Pyramid Height ( meters) 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 a) Write an expression that represents the volume of a pyramid. b ) Find the volume of the Great Pyramid of Cheops

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MISSION ACCMPLISHED MISSION ACCMPLISHED a) V=1/3 Bh b) V=4050(147) 3 V=198,450 cu. ft. V=198,450 cu. ft.

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FYI: In 1995, Sheila Hudson Strudwick set the U.S. women’s 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.

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

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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?

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MISSION ACCOMPLISHED MISSION ACCOMPLISHED 1 st change 2132ft 2 ½” to inches 2132*12 = 25584 + 2 ½” + 2 ½” 25,586.5” 25,586.5” Then divide by ½: 25,586.5/.5=51,173 25,586.5/.5=51,173 *There would have been 51,173 slices of bread.

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

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Mission Accomplished V=(h/3)(a^2+ab+b^2)V=(6/3)(2^2+2(4)+4^2)V=2(4+8+16)V=2(28)V=56

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MAT 105 SPRING 2009 Chapter 2 Geometry.

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