s A 10.) Solve by the method of false position: A quantity and its 2/3 are added together and from the sum 1/3 of the sum is subtracted, and 10 remains. What is the quantity? (problem 28 of the rind mathematical papyrus) 16.) Some scholars have conjectured that the area calculated in problem 10 of the Moscow Mathematical Papyrus is that of a semicylinder rather than a hemisphere. Show that the calculation in that problemdoes give the correct surface area of a semicylinder of diameter and height both equal to 4 1/2. 28.) Solve the problem from the old Babylonian tablet BM 13901: The sum of the areas of two squares is 1525. The side of the second square is 2/3 that of the first plus 5. Find the sides of each square. 34.) Solve the following problem from tablet YBC 6967: A number exceeds its reciprocal by 7. Find the number and reciprocal. (In this case, the two numbers are \"reciprocals\" means that their product is 60.) 38.) Given a circle of circumference 60 and a chord of length 12, what is the perpendicular distance from the chord to the circumference? (This problem is from tablet BM 85194) Questions B 2.)Represent 8/9 as a sum of distinct unit fractions. Express the result in the Greek notation. Note that the answer to this problem is not unique. 8.)Show that the nth triangular number is represented algebraically as Tn= n(n+1) / 2 and therefore that an oblong number is double a triangular number. 10.)Show using dots that eight times any triangular number plus 1 makes a square. Conversely, show that any odd square diminished by 1 becomes eight times a triangular number. Show these results algebraically as well. 12.)Construct five Pythagorean triples using the formula (n, n2-1/2, n2+1 / 2), where n is odd. Construct five different ones using the formula (m, (m/2)2-1, (m/2)2+1), where m is even. 20. Give an example of each of the four rules of inference discussed in the text. Question C 6. Prove proposition I-32, that the three interior angles of any triangle are equal to two right angles. Show that the proof depends on I-29 and therefore on postulate 5. 12. Prove proposition III-3, that if a diameter of a circle bisects a chord, then it is perpendicular to the chord. And if a diameter is perpendicular to a chord, then it bisects the chord. 18.Prove that the last nonzero remainder in the Euclidean algorithm applied to the numbers a, b, is in fact the greatest common divisor of a and b. 32. Find the two mean proportional between two cubes guaranteed by Proposition VIII-12. 36.Use Euclid's criterion in Proposition IX-36 to find the next perfect number after 8128. 40.Solve the equations of Proposition 86 of the Data algebraically. Show that the two hyperbolas defined by the equations each have their axes as the asymptotes of the other. \f\f