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Problem 1: Let T be the tetrahedron with vertices (0, 0, 0), (2,0,0), (0, 3, 0), and (0,0, 4) sketched below: 3 In this problem, you will use the method developed in §6.2 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 x of a solid body extending from r = a to x = b, then the Volume of the solid dr. (b) Notice that an arbitrary cross-section perpendicular to the x-axis has been drawn in the tetrahedron. The cross-section perpendicular to the x-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 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 x only. (Hint: Draw the face of the tetrahedron that lies in the cz-plane, along with the dashed line that represents the height of the cross-section triangle.) (d) What is the area A(x) 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 c 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)?

Problem 1: Let T be the tetrahedron with vertices (0, 0, 0), (2,0,0), (0, 3, 0), and (0,0, 4) sketched below: 3 In this problem, you will use the method developed in §6.2 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 x of a solid body extending from r = a to x = b, then the Volume of the solid dr. (b) Notice that an arbitrary cross-section perpendicular to the x-axis has been drawn in the tetrahedron. The cross-section perpendicular to the x-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 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 x only. (Hint: Draw the face of the tetrahedron that lies in the cz-plane, along with the dashed line that represents the height of the cross-section triangle.) (d) What is the area A(x) 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 c 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)?

Algebra for College Students
Algebra for College Students
10th Edition
ISBN:9781285195780
Author:Jerome E. Kaufmann, Karen L. Schwitters
Publisher:Jerome E. Kaufmann, Karen L. Schwitters
Chapter13: Conic Sections
Section13.1: Circles
Problem 48PS
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Problem 1: Let T be the tetrahedron with vertices (0, 0, 0), (2,0,0), (0, 3, 0), and (0,0, 4) sketched below: 3 In this problem, you will use the method developed in §6.2 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 x of a solid body extending from r = a to x = b, then the Volume of the solid dr. (b) Notice that an arbitrary cross-section perpendicular to the x-axis has been drawn in the tetrahedron. The cross-section perpendicular to the x-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 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 x only. (Hint: Draw the face of the tetrahedron that lies in the cz-plane, along with the dashed line that represents the height of the cross-section triangle.) (d) What is the area A(x) 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 c 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)?
Transcribed Image Text:Problem 1: Let T be the tetrahedron with vertices (0, 0, 0), (2,0,0), (0, 3, 0), and (0,0, 4) sketched below: 3 In this problem, you will use the method developed in §6.2 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 x of a solid body extending from r = a to x = b, then the Volume of the solid dr. (b) Notice that an arbitrary cross-section perpendicular to the x-axis has been drawn in the tetrahedron. The cross-section perpendicular to the x-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 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 x only. (Hint: Draw the face of the tetrahedron that lies in the cz-plane, along with the dashed line that represents the height of the cross-section triangle.) (d) What is the area A(x) 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 c 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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