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Additional Mathematics Project Work 2 2012 SELANGOR EDUCATION DEPARTMENT ADDITIONAL MATHEMATICS PROJECT WORK 2/2012 PART 1 (a) Connecting a series of two points on straight lines which cross over each other can create abstract geometric patterns. Such attractive and sophisticated designs are also known as “string art patterns”. They can be produced and created using geometrical instruments. Diagram 1 shows one of the designs which is created by drawing straight lines on the Cartesian plane. Diagram 1 1 Collect five such patterns from magazines, newspapers or internet. You can also design your own patterns. (b) Rene Descartes and Hero (or Heron) of Alexandria were two great mathematicians who had made great contributions in mathematics. Conduct a research and write on the history of the two mathematicians and their important contributions. Selangor Education Department Page 1 Additional Mathematics Project Work 2 2012 PART 2 Encik Ali owns a rectangular piece of land with a dimension of 3 km x 8 km. He wants to use part of the land for rearing poultry. (a) There is a small building near the poultry farm. Find out the outline of the building by drawing all the lines and curve below on the same axes. Name the type of the building. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. x = 8, Selangor Education Department Page 2 Additional Mathematics Project Work 2 2012 (b) Diagram 2 shows the dimensions of the land, PQRS and the shaded triangle ABC as the part of the land which he wants to use for rearing poultry. 1.5km S R N 3.5 km B C 8 km 1 km A 5 km 3 km P Q 3 km Diagram 2 Find the length of fence needed to fence up the poultry farm. Give your answer correct to the nearest km. (c) His friend, Encik Muthu, suggested that he should set up the poultry farm in the shape of a right-angled triangle. Encik Ali has agreed and decided to fence up the new triangle, ABD, with a right angle at B such that the area of triangle ABD must be equal to that of triangle ABC. By using two different methods, determine the position of D. (d) Encik Ali found that usually three chickens will lay three eggs in one and a half day. How many eggs will nine chickens lay in ten days? Selangor Education Department Page 3 Additional Mathematics Project Work 2 2012 PART 3 In order to increase his income, Encik Ali has decided to build a nursery on another part of his land. He bought 100 meters of fencing to fence another triangular area. He prefers to fence an isosceles triangle with the values of the measurement of area and length of sides, in integer. (a) Using calculator or ICT, help Encik Ali to explore the dimensions and area of the different isosceles triangles that can be formed by completing the table below. Hence, state the dimensions of the four isosceles triangles with areas and length of sides, in integer. (You may use Heron’s formula to find the area of a triangle.) [Heron’s Formula: Area of ABC = , where and a, b and c are the sides of ABC] a b c Area . . . . . . . . . . . . (b) Using calculus, show that the area of the triangle ABC with a perimeter of 100 meters is maximum when the triangle is an equilateral triangle. Hence, find the maximum area of the triangle ABC. FURTHER EXPLORATION Heron's Formula can be used to find the area of a triangle when the lengths of the three sides are known. There are many proofs of Heron's Formula. Most can be categorized as algebraic, geometric, or trigonometric. Conduct a research using the internet to find out one proof of the Heron’s Formula in any one of the list categories. Selangor Education Department Page 4 Additional Mathematics Project Work 2 2012 REFLECTION While you were conducting the project, what have you learnt? What moral values did you practice? Represent your opinions or feelings creatively through usage of symbols, illustrations, drawings or even in a song. END OF PROJECT PAPER Selangor Education Department Page 5