Final answer:
The temperature does not change with time throughout the entire homogeneous and infinite body as the temperature expression is independent of time. The entire body represents the region of constant temperature, and technically, it has no specific geometric shape because it is infinite. A finite portion of this body could be considered a parallelepiped or rectangular prism to exemplify the concept.
Step-by-step explanation:
The question is asking for the region where the temperature within a homogeneous, infinite body does not change with time. This temperature is given by the expression t(x,y,z,t) = x^4 y^4 – 6z^2 3xy – 6yz. To find the region where temperature does not change with time, we need to identify the terms in the equation that are independent of the variable t. The given expression does not explicitly include the variable t, which means the entire temperature function is independent of time. Therefore, the temperature remains constant throughout the body regardless of the time at any given location (x, y, z). Given the equation does not vary with time at any point in the space, the region in question is the entire body.
As for the geometric shape of this region, the question specifies that the body is infinite and homogeneous, indicating that the shape extends indefinitely in all directions, and thus does not have a standard geometric form. However, if we were to conceptualize a finite portion of this infinite body, we could say that any subsection we take would be a parallelepiped or a rectangular prism, since these are the simplest forms to carve out of a continuous uniform space.