To understand how the stress in the legs of a man who grows into a giant changes, we first need to consider how his weight and the cross-sectional area of his legs will change if his height increases by a factor of 8 , while his density remains constant.
Let's denote the original height of the man by
H, and his new height by
H′. Given that
H′=8H, we can say that the man's linear dimensions all increase by a factor of 8 .
Assuming the man's shape and density remain constant, his volume will scale with the cube of the change in his linear dimensions. Since his volume is directly proportional to his weight (assuming density is constant), his weight
W′ will increase by the cube of the linear scale factor:
W′=83W where
W is his original weight. This is because volume (and thus weight, given constant density) scales as the cube of height, and if height increases by a factor of 8 , volume and weight increase by
83.
Now, let's consider the stress in the legs, knowing that stress is defined as force (weight, in this context) per unit area. The critical area here is the cross-section of the man's legs.
The cross-sectional area
A of his legs will increase as the square of the linear dimension, since area scales with the square of the length. So, if his height increases by a factor of 8 , the cross-sectional area of his legs will increase by
82, i.e.,
A′=82A Stress is defined as the force divided by the area over which the force is applied (
σ=‌ ). The stress in his legs
σ′ after the growth could thus be given by:
σ′=‌Substituting the expressions for
W′ and
A′ gives:
σ′=‌=8‌So, the stress in the man's legs would increase by a factor of 8 after his growth, assuming his density and shape remain constant.
Therefore, the correct answer is:
Option D - 8