66.4k views
2 votes
Find the volume v of the described solid s. the base of s is the triangular region with vertices (0, 0), (5, 0), and (0, 5). cross-sections perpendicular to the y-axis are equilateral triangles.

2 Answers

4 votes

Final answer:

To find the volume of the solid with a triangular base and equilateral triangular cross-sections, we integrate the area of the cross-sections along the y-axis from 0 to 5 based on the changing side length, which is a linear function of y.

Step-by-step explanation:

The problem requires finding the volume of a solid with a triangular base and equilateral triangular cross-sections perpendicular to the y-axis. The base of the solid, S, is a right triangle with vertices at (0, 0), (5, 0), and (0, 5). The height of each equilateral triangular cross-section at any point y is the distance from the point (y, 0) to the hypotenuse of the triangular base, which is linear and can be described by the equation x + y = 5. Because cross-sections are equilateral triangles, each side of a cross-section is twice the length of the height. Therefore, the area of a cross-section is ½∙height²∙√3.

As we move along the y-axis from 0 to 5, the length of a side of each cross-section decreases linearly from 5 to 0. We can integrate the area of these cross-sections from y = 0 to y = 5 to find the total volume of the solid. The integral for the calculation of volume is:

∫ (2∙(5-y))² × √3/4 dy from y=0 to y=5

Upon integrating, this gives us the volume of the solid S.

User Alotor
by
7.4k points
2 votes
Find the volume v of the described solid s. the base of s is the triangular region with vertices (0, 0), (5, 0), and (0, 5). cross-sections perpendicular to the y-axis are equilateral triangles.
User Onovar
by
8.1k points

No related questions found

Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories