Answer:
The volume of the tetrahedron is:
[tex]\frac{50}{3}=16.667[/tex]
Step-by-step explanation:
The volume of the tetrahedron is given by the intersection of the planes x = 0, y = 0, z = 0 and the plane formed by the three points given.
The equation of the plane formed by the three points is:
Points: (2,0,0);(0,5,0);(0,0,4)
[tex]\pi :\frac{x}{2} +\frac{y}{5} +\frac{z}{4} =1[/tex]
It can also be expressed as:
[tex]10x + 4y+5z=20[/tex]
We have to calculate the triple integral, therefore we must define the domain:
The values of x are given by:
0≤x≤2
We will integrate the values of y between the y = 0 axis and the line formed when z = 0:
z=0 ⇒ 10x + 4y=2 ⇒ [tex]y=\frac{20-10x}{4} =[/tex]
[tex]0\leq y \leq 5-\frac{5}{2}x[/tex]
We will integrate the values of z between the plane z = 0 and the plane 10x + 4y+5z=20
10x + 4y+5z=20 ⇒ [tex]z=\frac{20-10x-4y}{5} =4-2x-\frac{4}{5}y[/tex]
[tex]0\leq z \leq 4-2x-\frac{4}{5}y[/tex]
The volume of the tetrahedron is:
[tex]\int\limits^{2}_{0} \ \ \ \int\limits^{5-\frac{5}{2} x}_{0} \ \ \ \int\limits^{4-2x-\frac{4}{5}y }_{0} {z} \, dz\, dy \,dx[/tex]
[tex]\int\limits^{2}_{0} \ \ \ \int\limits^{5-\frac{5}{2} x}_{0} \frac{z^2}{2}|^{4-2x-\frac{4}{5}y}_0\, dy \,dx[/tex]
[tex]\frac{z^2}{2}|^{z=4-2x-\frac{4}{5}y}_{z=0}=\frac{5}{25}(5x+2(y-5))^2[/tex]
[tex]\int\limits^{2}_{0} \ \ \ \int\limits^{5-\frac{5}{2} x}_{0} {\frac{5}{25}(5x+2(y-5))^2}\, dy \,dx[/tex]
[tex]\int\limits^{2}_{0} {-\frac{25}{6}(x-2)^3} dx=-\frac{25}{24} (x-2)^4|^{2}_{0}=\frac{50}{3}[/tex]
