Homework Help

Find the volume V of the described solid S. The base of S is the triangular region with...

user profile pic

user6978788 | Student, Undergraduate | Honors

Posted March 1, 2013 at 7:59 AM via web

dislike 1 like

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.

Tagged with calculus, integral, math, volume

1 Answer | Add Yours

Top Answer

user profile pic

sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted March 1, 2013 at 6:26 PM (Answer #1)

dislike 2 like

You need to notice that the base of the solid is a triangle that is bounded by x axis, y axis and the line `x = -y + 5` .

You need to integrate the function `x = -y + 5` with respect to y, using disk method, hence, you need to evaluate the area of cross section, such that:

`A = ((5 - y)(5 - y)*sin 60^o)/2 => A = (sqrt3)/4*(5 - y)^2`

You need to evaluate the volume such that:

`V = (sqrt3)/4*int_0^5 (5 - y)^2 dy`

You should come up with the substitution, such that:

`5 - y = t => -dy = dt => dy = -dt`

Changing the limits of integration and variables, yields:

`V = (-sqrt3)/4*int_5^0 (t)^2 dt => V = (sqrt3)/4*int_0^5 (t)^2 dt`

`V = (sqrt3)/4*t^3/3|_0^5`

Using the fundamental theorem of calculus, yields:

`V = (sqrt3)/4*(5^3/3 - 0^3/3) = > V = 125sqrt3/12`

Hence, evaluating the volume of solid S, under the given conditions, yields `V = 125sqrt3/12` .

Join to answer this question

Join a community of thousands of dedicated teachers and students.

Join eNotes