Find the volume of the tetrahedron whose vertices are the given points: (0,0,0), (2,0,0), (0,2,0), (0,0,2).

Matias Aguirre 2022-07-16 Answered
Finding the volume of the tetrahedron with vertices (0,0,0), (2,0,0), (0,2,0), (0,0,2). I get 8; answer is 4/3.
In this case, the tetrahedron is a parallelepiped object. If the bounds of such an object is given by the vectors A, B and C then the area of the object is A ( B × C ). Let V be the volume we are trying to find.
x 2 = 6 y 2 z 2 A = ( 2 , 0 , 0 ) ( 0 , 0 , 0 ) = ( 2 , 0 , 0 ) B = ( 0 , 2 , 0 ) ( 0 , 0 , 0 ) = ( 0 , 2 , 0 ) C = ( 0 , 0 , 2 ) ( 0 , 0 , 0 ) = ( 0 , 0 , 2 ) V = | a 1 a 2 a 3 b 1 b 2 b 3 c 1 c 2 c 3 | = | 2 0 0 0 2 0 0 0 2 | = 2 | 2 0 0 2 | = 2 ( 4 0 ) = 8
However, the book gets 4 3
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Answers (2)

Kendrick Jacobs
Answered 2022-07-17 Author has 16 answers
Explanation:
Note that the given volume is a cone with the height 2 and a right isosceles triangle of side 2 as the base. Thus, its volume can be calculated as 1 3 A r e a b a s e H e i g h t = 1 3 ( 1 2 2 2 ) 2 = 4 3 .
Did you like this example?
Subscribe for all access
Lorena Lester
Answered 2022-07-18 Author has 2 answers
Explanation:
Your tetrahedron is also a pyramid. with the volume of 1 3 2 2 2 2 = 4 3
Did you like this example?
Subscribe for all access

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

You might be interested in

asked 2022-07-14
Finding volume of solid in one quadrant - divide total volume by 4? 8? 2?
I want to find the volume of the solid produced by revolving the region enclosed by y = 4 x and y = x 3 in the first quadrant. The wording about the first quadrant confuses me but here's my work so far:
I know the volume unrestrained by quadrant is:
V = a b π ( f ( x ) 2 g ( x ) 2 ) d x
Where f ( x ) = 4 x and g ( x ) = x 3 . To find a and b, I look for the largest and smallest intersection points between the two functions:
f ( x ) = g ( x ) 4 x = x 3 0 = x 3 4 x = x ( x 2 ) ( x + 2 ) x { 2 , 0 , 2 }
Plugging all of these into the volume equation above:
V = a b π ( f ( x ) 2 g ( x ) 2 ) d x = 2 2 π ( ( 4 x ) 2 ( x 3 ) 2 ) d x = π 2 2 ( 16 x 2 x 6 ) d x = π ( 2 2 16 x 2 d x 2 2 x 6 d x ) = π ( [ 16 x 3 3 ] 2 2 [ x 7 7 ] 2 2 ) = π ( 16 ( 2 ) 3 3 16 ( 2 ) 3 3 ( 2 ) 7 7 ( 2 ) 7 7 ) = π ( 128 3 + 128 3 128 7 + 128 7 ) = π ( 256 3 256 7 ) = π ( 1024 21 ) = 1024 π 21
This is the volume for the entire function. I make an assumption that since I only want one quadrant and the function is symmetric about both the x- and y-axes, I simply divide it by four.
V whole = 1024 π 21 V one quadrant = 1024 π 21 × 1 4 = 256 π 21
I have no way of verifying my results. Can my assumption be made, or there's a differing method I should be using here?
If I'm now working in 3D space, would I instead divide it by eight? But if I'm revolving around x = 0, wouldn't the solid of revolution take four quadrants in 3D space, thus I should divide the total volume by 2?
asked 2022-08-05
Write S (surface area) of a cube interms of its volume, V. so far I have that s= 6a(for area) squared. and v = a 3
asked 2022-11-20
Finding the volume below the part of the plane which is above the xy-plane
My function is f ( x , y ) = 2 | x | | y | and I'm supposed to find the volume below the part of the plane which is above the xy-plane.
I don't understand how to find the limits of my integrals for this problem. I have tried to draw the lines for all the cases for which the absolute value of x and y is both positive and negative, but I don't see the limits. Is there anyone who has any suggestions?
asked 2022-09-24
Volume of a parallelepiped, given 8 vertices
Given the eight vertices (0,0,0), (3,0,0), (0,5,1), (3,5,1), (2,0,5), (5,0,5), (2,5,6), and (5,5,6), find the volume of the parallelepiped.
I'm having trouble finding the 1 vertex and 3 vectors needed to find the volume. The closest four vertexes I found so far are (0,0,0),(3,0,0),(0,5,1),(3,5,1)...is using those four vertexes correct? Any starting hints to point me in the right direction?
asked 2022-09-19
Finding volume of a sphere using integration
I have searched and found 2 methods of finding volume using integration:
- considering a small cylindrical element and integrating that over the radius
- considering a small circle element and using the relation x 2 + y 2 = r 2 and integrating it over the z-axis.
I was trying to find the integration by considering a small circle element (with radius r) and using the relation r = R cos θ where R is the radius of the sphere / hemisphere.
So I was thinking of calculating the volume of the hemisphere by integrating the π R 2 cos 2 θ d θ from θ to π / 2. Is this method right? And how will the integration be like?
asked 2022-11-03
Check my work? Finding line that divides rotational volume into equal parts. Not getting right answer.
I need to rotate the area between the curve y = x 2 and y = 9, bounded in the first quadrant, around the vertical line x = 3. I then must find the height (m) of the horizontal line that divides the resulting volume in half.
I've been trying to set up two integrals with washers. One from m 9 3 2 ( 3 x ) 2 d y and the other for the bottom region 0 m 3 2 ( 3 x ) 2 d y. The 3 is the outer radius of the washer and the 3 x gives the inner hole of the washer. I can't seem to get the right answer.
π m 9 ( 6 y y ) d y = π ( 4 y 3 2 1 2 y 2 ) evaluated from 9 to m = π ( 135 2 ( 4 m 3 2 1 2 m 2 ). Similarly, for the bottom region integral, I get π ( 4 m 3 2 1 2 m 2 ). I then try to set the volumes equal to each other and solve for m.
I believe the answer I should get is 9 2 3 but I don't get that value for m.
If I continue, I get 135 2 = 8 m 3 2 - m 2 but I don't see any easy way to solve without using a calculator.
asked 2022-10-22
Calculus Disk/Washer Method for Volume
I am given the bounded functions y = l n ( x ) , g ( x ) = .5 x + 3, and the x-axis. The reigon R is bounded between these, and I'm tasked with finding the volume of this solid using disk/washer method when revolved around the x-axis.
I know the formula I need to use, but I'm a little confused on finding the upper and lower limits and which to place as an innner and outer radius since the functions aren't graphed like the traditional washer method problem.