82. Find all the surfaces of a regular pentagonal prism, one side of a base being 8 feet, the apothem being 5.49+ feet, and the altitude of the prism 12 feet. 83. What is the height of a flag pole whose shadow is 5 feet when the shadow of an upright stick 4 feet long is 3 inches? 84. The length of a stick and the length of its shadow when placed upright are equal. At the same time what is the length of the shadow of a steeple 48 feet high? A 85. The line AD, which shows the altitude of the triangle ABC, when BC is con- E sidered the base, is 4 inches. BC is 6 inches, BE, the altitude, when AC is considered the D base, is 3 inches. Find AC. B 86. Arrange the following in three lists, one of lines, one of surfaces, the other of solids: Chord, circle, hexagon, octagon, cube, segment, arc, quadrilateral, sector, trapezium, pyramid, polygon, cone, perimeter, diameter, sphere, semicircle, circumference, radius, rectangle, quadrant, prism, liter, triangle, measure of an arc, measure of a sector, cylinder, semicircumference, decagon, rhombus, measure of a line, trapezoid, plane, diagonal, rhomboid, transversal, dodecagon, secant, median, square, parallelogram, tangent, parallelopiped. CHAPTER XIV. SQUARES AND CUBES. 1. Complete the table of the squares of the integers from 1 to 25 inclusive. 12 = 1, 22 = 4, etc. Make a table of the squares of numbers expressed by a significant figure and a cipher, as 10, 20, 30, and show how the second table is derived from the first. 2. Find with the help of your tables the side of a square whose area is 484 square meters. Of one whose area is 48,400 square meters. 3. Find the side of a square whose area is 529 square feet. Of one whose area is 52,900 square feet. 4. Find the length, width, and area of the rectangle formed by placing two squares, each containing 361 square inches, so that they have a common side. Illustrate. 5. Find the length of a rectangle twice as long as broad whose area is 1152 square inches. 6. Find the perimeter of a square whose area is 441 square meters. 7. Find the perimeter of a rectangle 3 times as long as broad whose area is 588 square meters. 8. What line squared and multiplied by 3 gives a rectangle containing 768 square inches? 9. If a represents the side of a square, what represents its area? The rectangle which contains three such squares? The width of the rectangle? Its length? 10. Find the width and length of a rectangle which is 5 times as long as wide, and contains 2205 square inches. 11. A rectangular lot whose length is 4 times its width contains 676 square rods. Find its dimensions. 12. How many yards of binding will it take for a square oilcloth mat covering 81 square feet? 13. Of two squares, the greater contains 24 square feet more than the less. The sum of their areas is 74 square feet. Find the side of each. SUGGESTION. - Let x = side of smaller square. x2 = area of smaller square, x2+ 24 = area of greater square. 14. The first of four squares contains 63 square inches more than the second, the second 17 square inches more than the third, the third 28 square inches more than the fourth. Their total area is 325 square inches. Find the side of each. 15. There are three squares, the first of which is 4 times the second, and the second is 9 times the third. Their combined area is 414 square feet. Find the length of their combined perimeters. 16. Find the number of rods of fencing required to inclose separately three square lots, the first of which contains 11 square rods more than the second, and the second contains 16 square rods more than the third, their combined area being 70 square rods. 17. Three squares are arranged as in the diagram. They cover 184 square inches. Find the length of the boundary line of the surface covered by them if the square on the left is 4 times as great as the middle square, and the middle square is 9 times as great as the one on the right. 18. The perimeter of a certain square is 4 inches longer than that of another square, and the sum of their perimeters is 100 inches. Find the sum of their squares. 19. How many times is the square of any number contained in the square of twice that number? Illustrate. 20. Draw squares, and show how many times the square of a 4-inch line is contained in the square of an 8-inch. In the square of a 12-inch line. 21. How many times is the square of any number contained in the square of 3 times that number? 22. How many times does the square of a line contain the square of of that line? 23. A square inch can be divided into how many figures each of an inch square? 24. Draw a figure which is the sum of the squares of two lines, one 8 inches, the other 3 inches, and find its area. 25. Draw a figure which is the square of the sum of the same lines, and find its area. 26. Find the difference between the sum of the squares and the square of the sum of two lines, one 7 inches, the other 8 inches. 27. Draw a rectangle which is the product of two lines respectively 8 inches and 5 inches, and find its area and perimeter. 28. Let AB and BC be two lines respectively 6 inches. and 4 inches, and AC their sum. How many square inches are there in the square ADEB? How many square inches are there in the square EHKF? How many square inches are there in the rectangle BEFC? How many square inches are there in the rectangle DGHE? How many in all? PRINCIPLE 51. The square of the sum of two lines is equal to the square of the first plus twice the product of the first by the second plus the square of the second. 29. Draw the square of the sum of two lines, one 5 inches, the other 3 inches, and show the truth of Prin. 51. 30. Square the sum of a and b, a representing a line 7 inches, and b representing a line 2 inches. = 31. Draw a figure, and show that (a + b)2 a2 + 2(ab)+ b2 when a a 10-inch line and b = a 3-inch line. 32. Let a and b be two lines. Show by numbers that the square of their sum is represented by the diagram. ax b a2 axb 33. x 10 inches, y = a line less than 10 inches, and the square of their sum = 289 inches. Find y. square 34. x = 10, y = a less number, the square of their sum = 324. Find y. square Find by trial the of the sum of two lines is y 36. The 256 square inches; the greater is 10 inches. Find the less. |