Answer:
Volume of the Tetrahedron T =
Step-by-step explanation:
As given, The tetrahedron T is bounded by the planes x + 2y + z = 2, x = 2y, x = 0, and z = 0
We have,
z = 0 and x + 2y + z = 2
⇒ z = 2 - x - 2y
∴ The limits of z are :
0 ≤ z ≤ 2 - x - 2y
Now, in the xy- plane , the equations becomes
x + 2y = 2 , x = 2y , x = 0 ( As in xy- plane , z = 0)
Firstly , we find the intersection between the lines x = 2y and x + 2y = 2
∴ we get
2y + 2y = 2
⇒4y = 2
⇒y = = 0.5
⇒x = 2() = 1
So, the intersection point is ( 1, 0.5)
As we have x = 0 and x = 1
∴ The limits of x are :
0 ≤ x ≤ 1
Also,
x = 2y
⇒y =
and x + 2y = 2
⇒2y = 2 - x
⇒y = 1 -
∴ The limits of y are :
≤ y ≤ 1 -
So, we get
Volume =
=
=
=
=
=
=
= 1 - 1² + - 0 + 0 - 0
= 1 - 1 + =
So, we get
Volume =