Answer
$6x+3y++2z=18$, and Volume of the cut off pyramid is $27$
Work Step by Step
As we are given that the equation of a plane is passing through the points $(1,2,3)$.
General form of equation of a plane is given as:
$p(x-1)+q(y-2)+r(z-3)=0$
If we write the equation of the plane in the form of intercepts as: $\dfrac{x}{p}+\dfrac{x}{q}+\dfrac{x}{r}=1$
The volume of the cut off pyramid is given as: $V=\dfrac{pqr}{6}$
Need to use Lagrange's Multiplier.
$f(x,y,z)=\lambda g(x,y,z)$
Take the partial derivatives, we get after solving
$p=3,q=6,r=9$
Thus, equation of a plane is given as:
$p(x-1)+q(y-2)+r(z-3)=0$
$6x+3y++2z=18$
Volume of the cut off pyramid is given as: $V=\dfrac{pqr}{6}=\dfrac{(3)(6)(9)}{6}$
Volume$=27 cm^3$
