EXAMPLES XVII 267 



(11) In the figure of Question 9, take P as the point of contact of 

 the bounding line of the figure and the axis of x. Use this point to 

 draw the first and second derived figures. Find the areas of these 

 figures, and use your results to find (1) the height of the centroid 

 above the axis of x, (2) the moment of inertia about the axis of x. 



(12) In the figure of Question 9, take Q as the point of contact of 

 the bounding line of the figure and the axis of y. Use this point to 

 draw the first and second derived figures. Find the areas of these 

 figures, and use your results to find (1) the distance of the centroid 

 from the axis of y, (2) the moment of inertia about the axis of y. 



(13) Draw a circle of 3 inches radius, and let PT be a tangent 

 to the circle, P being the point of contact. Using this point P, 

 draw the first and second derived figures. Find the areas of these 

 figures, and use your results to find the height of the centroid 

 above the tangent PT, and the moment of inertia about that tan- 

 gent. Verify your results. 



(14) The co-ordinates of five points A, B, C, D, and E are (1-5, 0), 

 (3-5, 0), (6, 3-5), (2, 6), and (0, 2-5) respectively, and these are the 

 five angular points of a polygon ABCDE. Plot the points and 

 draw the polygon. Let P be the mid-point of the side AB. Using 

 this point, draw the first and second derived figures, and find 

 their areas. Use your results to find the height of the centroid 

 of the polygon above the side AB and the moment of inertia about 

 the side AB. 



(15) The top of a reservoir is a rectangle of sides 2a and 26, 

 the depth is h, and the sides are inclined to the horizontal at 45. 

 Prove that the volume contained by the reservoir is 



(16) A wedge has a rectangular base 24 inches by 16 inches and 

 the height is 6 inches. The faces corresponding to the larger 

 sides of the base are inclined to the horizontal at 45, while those 

 corresponding to the smaller sides are inclined to the horizontal 

 at 60. Find the volume of the wedge. 



(17) In Question 16, what would be the height of the wedge so 

 that the top surface becomes a straight line ? What is the length 

 of this edge and what is the volume of the resulting wedge ? 



(18) The basis of Simpson's Rule is that if three successive 

 equidistant ordinates (distant h apart), y v y z , t/ 3 , are drawn to 

 any curve, the three points may be taken as lying on the curve 

 y = a + bx + ex*. Imagine y z to be the axis of y, so that ( h,y^), 

 (0, i/ 2 ), and (h, y 3 ) are the three points. Substitute these values 

 in the equation, and find a and c. Integrate a + bx + ex* between 

 the limits h and h and divide by 2h. This gives the average 

 value of y. Express it in terms of y lt y 2 , and t/ 3 . (B. of E., 1907.) 



