128 PRACTICAL MATHEMATICS 



(16) Find the dimensions of the cylinder of greatest curved 

 surface which can be inscribed in a sphere of 10 inches radius. 



(17) Find the dimensions of the cylinder of greatest total sur- 

 face which can be inscribed in a sphere of 10 inches radius. 



(18) Find the dimensions of a conical tent of greatest capacity, 

 the area of canvas used being 500 square yards. 



(19) If y = x* n x l+n where n = 0-885, for what value of x is 

 y a maximum, and what is the maximum value ? 



s 3 



(20) The cost of a ship per hour is c where c = 8-21 + r^rr, * 



1200 



being the speed in knots. Express the total cost of a passage 

 of 3400 nautical miles in terms of s. What value of s will make 

 this total cost a minimum ? Show that at speeds 10 per cent, 

 greater or less than this, the total cost is not very much greater 

 than what it is at the best speed. (B. of E., 1911.) 



s 3 



(21) The cost of a ship per hour is fc where c = 4 + ,, s 



1000 



being the speed in knots relatively to the water. Going up a 

 river whose current is 5 knots, what is the speed which causes 

 least total cost of passage ? (B. of E., 1905.) 



(22) In Q. 21, if the ship is going down the river, what is the 

 speed which causes least total cost of passage ? 



(23) From a rectangular sheet of tin 12" x 10 // equal squares 

 are cut from each corner, and the remainder is formed into a 

 rectangular vessel. Find the length of the side of the square 

 so that the volume of the vessel shall be greatest. 



(24) ABCD is a sheet of tin 10" square. From the corners A 

 and B squares of x" side are cut away, and from the corners C 

 and D rectangles of breadth x" are cut away. The remainder is 

 formed into a rectangular vessel with a lid. Find the dimension 

 x so that the volume of this vessel shall be greatest. 



(25) The stiffness of a beam of rectangular cross-section varies 

 as bh 3 where b is the breadth and h is the depth of a cross-section. 

 Find the dimensions of the stiffest beam which can be cut from 

 a cylindrical log of 24 inches diameter. 



(26) If y = ae- w sin (pt - c), and c = 0-135, a = 4, k = 300, 

 p = 500, find the first maximum and the first minimum value 

 of y, and the values of t which produce them. 



(27) Find the dimensions of the cylinder of greatest volume 

 which can be inscribed in a cone 5" high, radius of base 2". 



(28) Find the dimensions of the cylinder of greatest curved 

 surface which can be inscribed in a cone 5" high, radius of 

 base 2". 



(29) Find the dimensions of the cylinder of greatest total 

 surface which can be inscribed in a cone 5" high, radius of 

 base 2'". 



