Final answer:
To maintain the same volume when the diameter of a wire's cross-section decreases by 5%, the length must increase by approximately 10.53%. Therefore, the closest answer choice is 10%, which is option (b).
Step-by-step explanation:
The question involves understanding how changes in the dimensions of an object affect its volume when the material remains constant. Specifically, the student is asking if the diameter of the cross-section of a wire is decreased by 5%, how much the length must be increased to keep the volume the same.
In the case of a cylinder (which a wire can be modeled as), the volume V is given by the formula V = πr^2h, where r is the radius and h is the height or length of the cylinder. If the diameter (and thus the radius) decreases by 5%, the new radius becomes 0.95r. For the volume to remain constant (V' = V), the new length h' must compensate for the reduced cross-sectional area.
Therefore, the equation to solve is: π(0.95r)^2h' = πr^2h. Simplifying this equation, we find that h'/h = 1/(0.95^2). When calculated, this ratio turns out to be approximately 1.1053, meaning that the length must increase by about 10.53% to maintain the same volume when the radius decreases by 5%.
Hence, the correct answer is approximately 10%, so answer choice (b) is the closest correct option.