EXAMPLE. 223 



To find the value of y which makes this a minimum, we 

 must equate to zero its derivative with respect to y, which 

 gives 



*- gsin6> m 



-2-COS0' 



a sin 6 



Hence, for a given angle of slope 6, and for a given area 

 a, the trapezoidal section of least resistance is determined by 

 (2) and (5). 



Consequently, the width CD of the top is 



CD = x + 2y cot 6 



= - + y cot 0; (6) 



u 



v 

 and the value of , from (3), is 



Cv 



p 1 2 cos 



_ _ _ I __ ni 



a ~~ y a sin y 



= ? [from (4)]. (7) 



y 



EXAMPLE. 



What dimensions should be given to the transverse sec- 

 tion of a canal, when the angle of slope of its banks is to be 

 40, and when it is to carry 75 cubic feet of water with a 

 mean velocity of 3 feet ? 



Here we have 



a = ^ 25 square feet; 



and hence, from (5), we have the depth 



/ 25 sin 40 /0.64279 



y = V 2--^oT40 = V W3806 = 



