In Exercises 53 and 54, use a theorem of Pappus to find the centroid of the given triangle. Use the fact that the volume of a cone of radius r and height h is V = Trh. 53. 54. (a, c) 54. (0, b) (a, b) (0, 0) (0, 0) (a. 0) х
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- Which lines or line segments or rays must be drawn or constructed in a triangle to locate its a orthocenter? b centroid?A frustum of a cone is the portion of the cone bounded between the circular base and a plane parallel to the base. With dimensions are indicated, show that the volume of the frustum of the cone is V=13R2H13rh2Does the sphere x2+y2+z2=100 have symmetry with respect to the a x-axis? b xy-plane?
- Find the exact lateral area of each solid in Exercise 40. Find the exact volume of the solid formed when the region bounded in Quadrant I by the axes and the lines x = 9 and y = 5 is revolved about the a) x-axis b) y-axisUse cylindrical shells to find the volume of the cone generated when the triangle vertices (0,0), (0,r), (h,0), where r>0 and h>0, is revolved around the x-axis. Enter the exact answer.In Exercises 47–50, find the volume of the solid generated by revolv-ing each region about the y-axis. 47. The region enclosed by the triangle with vertices (1, 0), (2, 1), and (1, 1) 48. The region enclosed by the triangle with vertices (0, 1), (1, 0), and (1, 1) 49. The region in the first quadrant bounded above by the parabola y = x2, below by the x-axis, and on the right by the line x = 2 50. The region in the first quadrant bounded on the left by the circle x2 + y2 = 3, on the right by the line x = sqrt(3), and above by the line y = sqrt(3).
- In Exercises 31–32, find the volumes of the solids generated by revolving the regions about the given axes. If you think it would be better to use washers in any given instance, feel free to do so. 31. The triangle with vertices (1, 1), (1, 2), and (2, 2) about a. the x-axis b. the y-axis c. the line x = 10/3 d. the line y = 1 32. The region bounded by y = srqt(x), y = 2, x = 0 about a. the x-axis b. the y-axis c. the line x = 4 d. the line y = 2Use cylindrical shells to find the volume of the cone generated when the triangle with vertices (0, 0), (0, r), (h,0), where r > 0 and h > 0, is revolved about the x-axis.Use Pappus's Theorem to find the volume of the solid obtained when the region bounded by y=4x2, y=0, and x=1, whose centroid is 34,310, is revolved about the y-axis. Check your answer by using the method of cylindrical shells. What is the volume of the solid in cubic units?
- Use Pappus's theorem for surface area and the fact that the surface area of a sphere of radius q is 4πq2 to find the centroid of the semicircle y=q2−x2. What is the the centroid of the semicircle y=q2−x2 is x,y, what is x= and what is y=?Use the Theorem of Pappus to find the exact volume of a general cone created by rotating a triangle with vertices (0,0), (3,0), and (0,7) around the y-axis. Volume=Find the volume of the solid bounded below by the plane z = 2 and above by the sphere x^2+y^2+z^2=72.Write your answer as an exact value in terms of π. Notice a factor π is already included in the answer.Volume = π