Final answer:
When the area of a wire is doubled, the resistance is halved. If the same wire had its diameter doubled instead, the area would be quadrupled, decreasing the resistance to one quarter. Stretching the wire to thrice its length increases resistance nine-fold.
Step-by-step explanation:
When the cross-sectional area of a wire is doubled, the resistance of the wire is halved, according to the formula for the resistance of a conductor: R = ρL/A, where R is resistance, ρ (rho) is the electrical resistivity of the material, L is the length, and A is the cross-sectional area.
In the case of a wire of the same material with the same length but double the diameter, the cross-sectional area is quadrupled, since area is π(•d/2)^2. Thus, the resistance would be one quarter of the original wire.
If a wire is stretched to three times its original length without changing its volume, its cross-sectional area decreases, and thus the resistance would increase by a factor of nine, because resistance is proportional to length times the reciprocal of the cross-sectional area.