Volume of Solid with Elliptical Base and Isosceles Triangle Cross Sections

  • Thread starter mrdoe
  • Start date
  • Tags
    Volume
In summary, the base of the solid is in the form of an ellipse and the volume is found by taking the integral of the area of the cross sections perpendicular to the x-axis. The volume is approximately 11.781 when the axis is the x-axis and 7.54 when the axis is the y-axis. The correct equation for y is y = (4/5)sqrt(25-x^2) and the mistake was dividing by 4 instead of multiplying.
  • #1
mrdoe
36
0
Q: A solid has a base in the form of the ellipse: x^2/25 + y^2/16 = 1. Find the volume if every cross section perpendicular to the x-axis is an isosceles triangle whose altitude is 6 inches.
I got 11.781 (or 3.75pi) but I just wanted to check my answer.
Q: Use the same base and cross sections as #3, but change the axis to the y-axis.
Here I got 7.54 (or 2.4pi)
 
Physics news on Phys.org
  • #2
I think your answers are way too low. You'd better show your work.
 
  • #3
For the first one

[tex]
\frac{y^2}{16} = 1-\frac{x^2}{25}[/tex]

[tex]
y=\frac{\sqrt{25-x^2}}{20}[/tex]

[tex]
A=\frac{3\sqrt{25-x^2}}{10}[/tex]

[tex]
V=\int^5_{-5}Adx[/tex]

[tex]
V=\frac{3}{10}\left[\frac{1}{2}\sqrt{25-x^2}x + \frac{25}{2}\arcsin\left(\frac{x}{5}\right)\right]\left|^5_{-5}[/tex]

[tex]
\approx 11.781
[/tex]
 
  • #4
Making y the subject of the ellipse equation I got:

[tex]y=\pm \frac{4}{5}\sqrt{25-x^2}[/tex]

I can't seem to follow where you went wrong there.
 
  • #5
y^2/16=1-x^2/25. y=4*sqrt(1-x^2/25). y=(4/5)*sqrt(25-x^2). A=(1/2)*6*(2*y). Stuff like that. You are doing something wrong.
 
  • #6
EDIT: I accidentally divided by 4 on the other side instead of multiplying.
 
Last edited:

1. What is a simple volume problem?

A simple volume problem involves finding the volume of a regular geometric shape, such as a cube, cylinder, or sphere.

2. How do you calculate the volume of a cube?

The volume of a cube is calculated by multiplying the length, width, and height of the cube. The formula is V = l x w x h, where V represents the volume, l represents the length, w represents the width, and h represents the height.

3. What is the formula for finding the volume of a cylinder?

The formula for finding the volume of a cylinder is V = πr²h, where V represents the volume, r represents the radius, and h represents the height of the cylinder.

4. Can you use the same formula for finding the volume of a cone?

No, the formula for finding the volume of a cone is V = (1/3)πr²h, where V represents the volume, r represents the radius, and h represents the height of the cone. This formula takes into account the slanted sides of the cone.

5. How do you convert between different units of volume?

To convert between different units of volume, you can use conversion factors. For example, to convert from cubic centimeters (cm³) to liters (L), you would multiply the volume in cm³ by 0.001. It is important to make sure that the units are consistent when converting between different units of volume.

Similar threads

  • Calculus and Beyond Homework Help
Replies
4
Views
905
  • Calculus and Beyond Homework Help
Replies
10
Views
1K
  • Calculus and Beyond Homework Help
Replies
4
Views
697
  • Calculus and Beyond Homework Help
Replies
6
Views
5K
  • Calculus and Beyond Homework Help
Replies
2
Views
3K
Replies
2
Views
1K
  • Calculus and Beyond Homework Help
Replies
3
Views
1K
  • Calculus and Beyond Homework Help
Replies
3
Views
2K
  • Calculus and Beyond Homework Help
Replies
2
Views
7K
  • Calculus and Beyond Homework Help
Replies
2
Views
3K
Back
Top