In this problem, you will use the method developed in §6.1 to find the volume of the tetrahedron depicted in the above sketch. (a) Fill in the missing parts of the following statement: if A(x) is the area of the vertical cross-section at z of a solid body extending from r = a to r = b, then the Volume of the solid : |dr. (b) Notice that an arbitrary cross-section perpendicular to the r-axis has been drawn in the tetrahedron. The cross-section perpendicular to the r-axis is a triangle. We know the area of a triangle is A = bh. Let the line in the ry-plane denote the base of the triangle and the line in the rz-plane be the height of the crogs-section. Find a formula for the base of the cross-section triangle in terms of r only

In this problem, you will use the method developed in §6.1 to find the volume of the tetrahedron depicted in the above sketch. (a) Fill in the missing parts of the following statement: if A(x) is the area of the vertical cross-section at z of a solid body extending from r = a to r = b, then the Volume of the solid : |dr. (b) Notice that an arbitrary cross-section perpendicular to the r-axis has been drawn in the tetrahedron. The cross-section perpendicular to the r-axis is a triangle. We know the area of a triangle is A = bh. Let the line in the ry-plane denote the base of the triangle and the line in the rz-plane be the height of the crogs-section. Find a formula for the base of the cross-section triangle in terms of r only

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

1. Let T be the tetrahedron with vertices (0,0,0), (2, 0, 0), (0,3, 0), and (0,0, 4) sketched below:
In this problem, you will use the method developed in §6.1 to find the volume of the tetrahedron depicted
in the above sketch.
(a) Fill in the missing parts of the following statement: if A(x) is the area of the vertical cross-section
at a of a solid body extending from r = a to a = b, then the
Volume of the solid
dr.
(b) Notice that an arbitrary cross-section perpendicular to the r-axis has been drawn in the tetrahedron.
The cross-section perpendicular to the r-axis is a triangle. We know the area of a triangle is A = bh.
Let the line in the ry-plane denote the base of the triangle and the line in the az-plane be the height
of the cross-section. Find a formula for the base of the cross-section triangle in terms of r only.

Volume of the solid
dr.
(b) Notice that an arbitrary cross-section perpendicular to the r-axis has been drawn in the tetrahedron.
The cross-section perpendicular to the r-axis is a triangle. We know the area of a triangle is A = bh.
Let the line in the ry-plane denote the base of the triangle and the line in the xz-plane be the height
of the cross-section. Find a formula for the base of the cross-section triangle in terms of r only.
(Hint: Draw the face of the tetrahedron that lies in the ry-plane, along with the dashed line that
represents the base of the cross-section triangle.)
(c) Next find a formula for the height of the cross-section triangle in terms of r only. (Hint: Draw
the face of the tetrahedron that lies in the az-plane, along with the dashed line that represents the
height of the cross-section triangle.)
(d) What is the area A(r) of the cross-section drawn in the sketch?
(e) Use your area function and integration with respect to a to find the volume of the tetrahedron T.
(f) Now suppose the tetrahedron has vertices (0,0,0), (a,0, 0), (0, 6,0), and (0,0, c), where a, b, and e
are all positive numbers. Use integration with respect to r to find the volume of the tetrahedron.
(Your answer should be in terms of a, b, and c.) Does your formula agree with the volume you
found in part (e)?

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