Answer and Explanation:
Let $M$ be the mass of both plates and $A$ be their equal area.
The moment of inertia (MOI) of a uniform rectangular plate of mass $M$, length $L$, and width $W$ about an axis passing through its center and parallel to its length ($L$) is $I_W = \frac{1}{12} M W^2$. The MOI about an axis passing through its center and parallel to its width ($W$) is $I_L = \frac{1}{12} M L^2$.
For plate S (square):
Let the side length of the square plate S be $L_S$.
Its area is $A = L_S^2$, so $L_S = \sqrt{A}$.
The MOI about the x-axis (or y-axis) passing through its center is:
$I_{x,S} = I_{y,S} = \frac{1}{12} M L_S^2 = \frac{1}{12} M A$.
For plate R (rectangular):
Let the length of the rectangular plate R be $L_R$ and its width be $W_R$.
Its area is $A = L_R W_R$.
From the diagram, the x-axis is aligned with the longer dimension ($L_R$) and the y-axis is aligned with the shorter dimension ($W_R$). Therefore, $L_R > W_R$.
The MOI about the x-axis (parallel to $L_R$) is:
$I_{x,R} = \frac{1}{12} M W_R^2$.
The MOI about the y-axis (parallel to $W_R$) is:
$I_{y,R} = \frac{1}{12} M L_R^2$.
Comparison of MOIs:
Since $L_R W_R = A$ and $L_R > W_R$, it implies that $L_R > \sqrt{A}$ and $W_R < \sqrt{A}$.
(If $L_R = \sqrt{A}$, then $W_R = \sqrt{A}$, making it a square. If $L_R$ is greater than $\sqrt{A}$, then $W_R$ must be less than $\sqrt{A}$ for their product to be $A$.)
1. Comparing MOI about the x-axis:
$I_{x,R} = \frac{1}{12} M W_R^2$
$I_{x,S} = \frac{1}{12} M A$
Since $W_R < \sqrt{A}$, it follows that $W_R^2 < A$.
Therefore, $I_{x,R} < I_{x,S}$.
This means statement b. x-axis for R is less than that for S is correct.
2. Comparing MOI about the y-axis:
$I_{y,R} = \frac{1}{12} M L_R^2$
$I_{y,S} = \frac{1}{12} M A$
Since $L_R > \sqrt{A}$, it follows that $L_R^2 > A$.
Therefore, $I_{y,R} > I_{y,S}$.
This means statement c. y-axis for R is greater than that for S is correct.
Conclusion:
Statements b and c are correct.
Let's check the given options:
a. x-axis for R is equal to that for S (False)
b. x-axis for R is less than that for S (True)
c. y-axis for R is greater than that for S (True)
d. y-axis for R is less than that for S (False)
The option that includes both correct statements is (2) both b & c.
The final answer is $\boxed{2}$
