214 . EXAMPLES OF MAXIMA AND MINIMA. 



30. A Norman window consists of a rectangle surmounted by 

 a semicircle. Given the perimeter, required the height 

 and breadth of the window when the quantity of light 

 admitted is a maximum. 



The radius of the semicircle must equal the height 

 of the rectangle. 



31 . Shew that the altitude of the greatest equilateral triangle 

 that can be circumscribed about a given triangle, is 



(a 1 + J a -2o& cos (TT +(?)}*. 



32. A .straight line is drawn through the given point P, 

 meeting the axes Ox and Oy at A and B respectively 

 (see the figure on page 198) ; find the position of the 

 straight line, 



(1) When AB is a minimum. 



(2) When OA + OB is a minimum. 



(3) When OA x OB is a minimum. 



(4) When OA + OB + AB is a minimum. 

 , (5) When OA x OB x AB is a minimum. 



(6) When OA" + OB" is a minimum. 



Let denote the angle PAO, then we must have 



(2) tan -(?)*, 



(3) 



(4) 



, 



a + V(2oi) ' 



(5) 2atan 8 0-6ta 

 i 



(I \n-f-l 

 a) ' 



33. Having given an angle of a triangle and the opposite 

 side, prove that the area will be a maximum when the 

 given angle is equidistant from the other angles. 



