Explanation:
To solve this problem, we need to understand how resistance is related to the dimensions of a rod.

Resistance of a Rod:
The resistance of a cylindrical rod is given by the formula:
R = ρ * (L / A),
where R is the resistance, ρ is the resistivity of the material, L is the length of the rod, and A is the cross-sectional area of the rod.

Volume of a Cylinder:
The volume of a cylinder is given by the formula:
V = π * r^2 * h,
where V is the volume, r is the radius of the cylinder, and h is the height (or length) of the cylinder.

Given Conditions:
1. The volume of the rod remains constant.
2. The length of the rod is reduced to half its original length.

Analysis:
Since the volume of the rod is constant, we can write:
π * r^2 * L = π * r^2 * L',
where L is the original length and L' is the new length.

We are asked to find the new resistance after the rod is reformed to half its original length while keeping the volume constant.

Solution:
Let's assume the original length of the rod is L, and its radius is r.
From the given conditions, we know that the new length of the rod is half its original length, i.e., L' = L/2.

Step 1:
We can rewrite the equation for the volume of the rod as:
π * r^2 * L = π * r^2 * (L/2).
Simplifying this equation, we find:
L = L/2.
This implies that the original length of the rod is equal to twice its new length.

Step 2:
Now, let's consider the resistance of the rod.
The resistance before the change is given by:
R = ρ * (L / A).

Step 3:
To find the resistance after the change, we need to determine the new length and the new cross-sectional area of the rod.

The new length of the rod is L/2, as given.

Step 4:
The cross-sectional area of a cylindrical rod is given by the formula:
A = π * r^2.

Step 5:
Let's substitute the new length (L/2) and the new cross-sectional area (π * r^2) into the resistance formula:
R' = ρ * (L/2) / (π * r^2).
Simplifying this equation, we find:
R' = (ρ * L) / (2 * π * r^2).

Step 6:
Now, let's substitute L = 2L' into the equation for R':
R' = (ρ * 2L') / (2 * π * r^2).
Simplifying this equation, we find:
R' = (ρ * L') / (π * r^2).

Step 7:
Comparing the expression for R' with the expression for R, we can see that:
R' = R / 4.

Therefore, the resistance after the reformation of the rod will be R/4.