What is the Volume of a Solid with a Triangular Base and Square Cross-Sections?

  • Thread starter Thread starter ProBasket
  • Start date Start date
  • Tags Tags
    Base Solid
Click For Summary

Homework Help Overview

The problem involves finding the volume of a solid that has a triangular base defined by the vertices at (-8,4), (4,4), and the origin, with square cross-sections perpendicular to the y-axis. The original poster expresses uncertainty about how to approach the problem after initially graphing the points.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • One participant suggests considering an element of size dy at a distance y from the origin and finding the width of this element in terms of y. They also mention that the height of the element is the same as the width due to the square cross-section. Another participant reiterates the problem statement and provides equations for the bounding lines of the triangle, leading to a calculation of the cross-sectional area and differential volume.

Discussion Status

The discussion includes attempts to clarify the setup of the problem and explore the relationships between the dimensions of the solid. Some guidance on integrating to find the volume has been provided, but there is no explicit consensus on the final approach or outcome.

Contextual Notes

Participants are working with the constraints of the problem as stated, including the specific vertices of the triangular base and the requirement for square cross-sections. There is an indication of varying height along the y-axis, which is a key aspect of the volume calculation.

ProBasket
Messages
140
Reaction score
0
The base of a certain solid is the triangle with vertices at (-8,4) ,(4,4), and the origin. Cross-sections perpendicular to the y-axis are squares.

Then the volume of the solid is __________

i have no clue or don't even know how to start this problem. Well the first thing i did was just graphed the three points they gave me, but that's about it. can someone lend me a hand?
 
Physics news on Phys.org
The height of this solid varies along the y axis. Cosider an element of size dy at a distance y from the origin along the y axis. Find the width of this element using the given information in terms of y. What is the height of the element. (same as the width since cross section is squre). So now you can find the volume of the element.
call this dv
Intergrate dv from y=0 to y= y1 where y1 is the distance from origin to the base of the triangle.
 
ProBasket said:
The base of a certain solid is the triangle with vertices at (-8,4) ,(4,4), and the origin. Cross-sections perpendicular to the y-axis are squares.

Then the volume of the solid is __________

i have no clue or don't even know how to start this problem. Well the first thing i did was just graphed the three points they gave me, but that's about it. can someone lend me a hand?
From problem statement:
{Base Vertices (0,0) & (-8,4)} ⇒ {Bounding Base Line} = {y = (-1/2)*x} = {x = (-2)*y}
{Base Vertices (0,0) & (4,4)} ⇒ {Bounding Base Line} = {y = x} = {x = y}
{Base Vertices (-8,4) & (4,4)} ⇒ {Bounding Base Line} = {y = 4}

{Cross-section Area ⊥ y-axis} = {Width}*{Height} =
= {y - (-2)*y}*{y - (-2)*y} = ::: (Square cross-section ⇒ height=width)
= {3y}*{3y} =
= 9*y2

{Differential Volume} = dV = {Cross-section Area}*dy = {9*y2}*dy

[tex]\ \ \ \ (Volume) \ = \ \int_{0}^{4} 9y^{2} dy \ = \ \left [ 3y^{3} \right ]_{0}^{4}[/tex]

[tex]\ \ \ \color{red} (Volume) \ = \ (192)[/tex]


~~
 
Last edited:
thanks so much
 

Similar threads

  • · Replies 2 ·
Replies
2
Views
2K
  • · Replies 4 ·
Replies
4
Views
2K
  • · Replies 14 ·
Replies
14
Views
3K
Replies
3
Views
2K
  • · Replies 4 ·
Replies
4
Views
8K
  • · Replies 4 ·
Replies
4
Views
3K
  • · Replies 2 ·
Replies
2
Views
8K
Replies
5
Views
2K
Replies
1
Views
3K
  • · Replies 4 ·
Replies
4
Views
2K