169. Find to the nearest square inch the quantity of leather required to cover a spherical foot-ball which measures 23 inches in circumference. 170. Find the volume of a cask in gallons, the length being 47.5 inches, the bung diameter 28 6, and the head diameter 26.5. 171. An area in the form of an equilateral triangle is paved at the rate of 9d. per square foot, and it is fenced at the rate of 5 shillings per foot: shew that the cost of fencing is to the cost of paving as 80/3 is to three times the number of feet in a side. 172. Find the side of an equilateral triangle, supposing it cost as much to pave the area at 9d. per square foot as to fence the three sides at 5 shillings per foot. 173. A rectangle is ·202 of an inch longer than a certain square, and 2 of an inch narrower; but contains the same area: shew by a figure that the number of inches in the side of the square is the product of 202 into '2, divided by their difference. 174. In measuring the edges of a cubical box to ascertain its content, an error of 202 of an inch is made in excess for the length, and of 2 of an inch in defect for the breadth, the height being properly measured; the calculated volume agrees with the true volume: find the volume in cubic inches. 175. The radius of a circle is 2 inches; two parallel straight lines are drawn in it, each an inch from the centre: find the area of the part of the circle between the straight lines. 176. A square hole 2 inches wide is cut through a solid cylinder of which the radius is 2 inches, so that the axis of the hole cuts at right angles the axis of the cylinder: find how much of the material is cut out. 177. A vessel is to be made in the form of a rectangular parallelepiped on a square base, and another vessel of the same capacity in the form of a right circular cylinder; the vessels are to have no lids, and each vessel is to be made of the most advantageous shape for the sake of saving material: shew that the material in the cylindrical vessel is about 92 of the material in the other vessel. 178. Shew that the same result as in the preceding Example holds if the vessels are to have lids. 179. A pyramid on a square base has every edge 100 feet long: find the edge of a cube of equal volume. 180. Verify by calculating various cases the following statement: a right circular cone is divided into a cone and a frustum of a cone, and the frustum is trimmed just enough to reduce it to a right circular cylinder; if the height of the frustum is one-third of the height of the original cone the volume of the cylinder is greater than in any other case, and is four-ninths of the volume of the original cone. |