How do you Find the volume of the solid that lies in the first octant and is bounded by the three coordinate planes and another plane passing through (3,0,0), (0,4,0), and (0,0,5)?

Question

Scarlett Allison · Accepted Answer

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The solid in question is a tetrahedron #V-ABC# having vertices #O(0,0,0), A(3,0,0), B(0,4,0) and C(0,0,5).# Its volume #V# is given by, #V=|1/6[vec(OA),vec(OB),vec(OC)]|,# where, #[vec(VA),vec(VB),vec(VC)]# denotes the scalar triple product #vec(VA)*{vec(VB)xxvec(VC)}.# We have, #vec(VA)=A(3,0,0)-O(0,0,0)=(3,0,0).# Similarly, #vec(OB)=(0,4,0), and vec(OC)=(0,0,5).# #:.[vec(VA),vec(VB),vec(VC)]=|(3,0,0),(0,4,0),(0,0,5)|,# #=3*4*5=60.# #rArr V=1/6|60|=10.#

Cameron Alexander · Answer

undefined To find the volume of the solid bounded by the three coordinate planes and the plane passing through the given points, you can use the formula for the volume of a tetrahedron, which is given by:

$$ V = \frac{1}{6} \left| \begin{array}{ccc} x_1 & y_1 & z_1 \ x_2 & y_2 & z_2 \ x_3 & y_3 & z_3 \end{array} \right| $$

where $ (x_1, y_1, z_1) $, $ (x_2, y_2, z_2) $, and $ (x_3, y_3, z_3) $ are the coordinates of the three vertices of the tetrahedron that are not on the coordinate planes, and the determinant is the absolute value of the determinant of the matrix formed by the coordinates.

In this case, the three vertices are (3,0,0), (0,4,0), and (0,0,5). Plugging these values into the formula, we get:

$$ V = \frac{1}{6} \left| \begin{array}{ccc} 3 & 0 & 0 \ 0 & 4 & 0 \ 0 & 0 & 5 \end{array} \right| $$

Calculating the determinant, we get:

$$ V = \frac{1}{6} \left| (3 \times 4 \times 5) - (0 \times 0 \times 0) - (0 \times 0 \times 0) \right| $$

$$ V = \frac{1}{6} \left| 60 \right| $$

$$ V = \frac{60}{6} $$

$$ V = 10 $$

So, the volume of the solid is 10 cubic units.

Elijah Abram · Answer

undefined To find the volume of the solid bounded by the three coordinate planes and the plane passing through the points (3, 0, 0), (0, 4, 0), and (0, 0, 5) in the first octant, you can use the formula for the volume of a tetrahedron.

First, find the equation of the plane passing through the given points. You can do this using the formula for the equation of a plane given three non-collinear points.

Once you have the equation of the plane, determine the points where it intersects each of the coordinate planes. These points will define the vertices of the tetrahedron.

Next, calculate the distances between the vertices to find the lengths of the edges of the tetrahedron.

Finally, use the formula for the volume of a tetrahedron, which is one-third of the base area multiplied by the height, to find the volume of the solid bounded by the coordinate planes and the plane passing through the given points.

Alternatively, you can also use triple integration to find the volume of the solid. Define the region of integration in terms of the given points and the coordinate axes, set up the triple integral, and evaluate it to find the volume.