A tetrahedron is formed in the �rst quadrant by the three coordinate

planes and the plane that meets the x-axis at a > 0, the y-axis at
b > 0 and the z-axis at c > 0. Compute its volume.
Kindly help

first quadrant

you can find the area B of the base (a triangle), and its height (the z-intercept)

v = 1/3 Bh

To compute the volume of a tetrahedron, we can use the formula:

Volume = (1/6) * base_area * height

The base area can be calculated as the area of a triangle formed by the three intersecting coordinate planes. We can find the length of the base by calculating the distance between the points where the plane intersects each axis.

Let's calculate the coordinates of the three points:

Point A: (a, 0, 0)
Point B: (0, b, 0)
Point C: (0, 0, c)

Now, let's calculate the distance between these points:

AB = sqrt((0 - a)^2 + (b - 0)^2 + (0 - 0)^2) = sqrt(a^2 + b^2)
AC = sqrt((0 - a)^2 + (0 - 0)^2 + (c - 0)^2) = sqrt(a^2 + c^2)
BC = sqrt((0 - 0)^2 + (b - 0)^2 + (c - 0)^2) = sqrt(b^2 + c^2)

Next, let's calculate the base area using Heron's formula:

s = (AB + AC + BC) / 2 (semi-perimeter)
base_area = sqrt(s * (s - AB) * (s - AC) * (s - BC))

Finally, calculate the height of the tetrahedron, which is the distance between the origin (0, 0, 0) and the plane:

height = |Ax + By + Cz + D| / sqrt(A^2 + B^2 + C^2)

Where A, B, C, and D are the coefficients of the equation of the plane in the form Ax + By + Cz + D = 0.

In this case, since the plane passes through the origin (0, 0, 0), the equation of the plane is:

Ax + By + Cz = 0

And the coefficients are:
A = 1
B = 1
C = 1

Therefore, the height can be computed as:

height = |a + b + c| / sqrt(1^2 + 1^2 + 1^2) = |a + b + c| / sqrt(3)

Now that we have all the necessary values, we can calculate the volume using the formula mentioned earlier:

Volume = (1/6) * base_area * height

Plug in the values for base_area and height to find the volume of the tetrahedron in the first quadrant.