Solve the problem.
-Find the moment of inertia I $L$ of the rectangular solid of density $\delta ( x , y , z ) = 1$ defined by $0 \leq x \leq 8,0 \leq y \leq 3$, $0 \leq z \leq 7$, where $L$ is the line through the points $( 2,1,0 ) , ( 2,2,0 )$.
A) 4312
B) 5208
C) 1624
D) 6104

Question

Quizplus · Accepted Answer

First, we need to find the equation of line $L$. Since the line passes through $(2,1,0)$ and $(2,2,0)$, we know that the $x$-coordinate is constant and equal to $2$. Thus, the equation of $L$ is given by $x=2$.

To find the moment of inertia, we need to integrate the density function $\delta(x,y,z)=1$ over the rectangular solid, weighted by the distance of each infinitesimal mass element from the axis of rotation (which is $L$ in this case) squared. The moment of inertia is then given by:

$$
I(L) = \iiint_E \delta(x,y,z)(	ext{distance from }L)^2\,dV,
$$

where $E$ is the rectangular solid.

Since $x=2$ lies on $L$, we can rewrite this as:

$$
I(L) = \int_{0}^{7}\int_{0}^{3}\int_{2}^{2} (	ext{distance from }L)^2\,dx\,dy\,dz = 0.
$$

Thus, the moment of inertia is $0$ and none of the answer choices are correct. This may be because there is a mistake in the problem statement or in the answer choices.

Solve the Problem LLL Of the Rectangular Solid of Density δ(x,y,z)=1\delta ( x , y , z ) = 1δ(x,y,z)=1

Solve the Problem $L$ Of the Rectangular Solid of Density $\delta ( x , y , z ) = 1$