The volume of a rectangular prism is given by the formula:
V = l × w × h
where l, w, and h are the length, width, and height of the prism, respectively.
Let's assume that the original dimensions of the prism are l₁, w₁, and h₁, and the new width after the scale factor is applied is w₂ = (1/3)w₁.
The new volume of the prism is given by:
V₂ = l₁ × w₂ × h₁
Substituting w₂ = (1/3)w₁, we get:
V₂ = l₁ × (1/3)w₁ × h₁
V₂ = (1/3)l₁w₁h₁
The new volume is 1/3 of the original volume. Therefore, if the width of the rectangular prism is changed by a scale factor of 1/3, the volume of the same shape will decrease by a scale factor of 1/3, and the new volume will be:
V₂ = (1/3) × 768 ft³ = 256 ft³
Therefore, the volume of the same shape if the width is changed by a scale factor of 1/3 is 256 ft³.