Final answer:
A property that always stays the same is the mass of an object, and dimensional analysis helps confirm consistent units in formulas. Volume and cross-sectional area are scalar quantities that can change with temperature, but the mass of an object remains constant unless material is added or removed. The correct answer is b) Mass of an Object.
Step-by-step explanation:
The question seems to be asking about dimensional consistency in geometric formulas, scalar quantities, and the properties of an object that remain constant. An example of a property that always stays the same is the mass of an object, assuming it is not subjected to any process that adds or removes mass.
When dealing with geometry and changes in dimensions due to temperature, a crucial concept is understanding how different properties like volume and cross-sectional area will change.
Regarding dimensional analysis in geometric formulas, for a formula to be dimensionally consistent, the dimensions on both sides of the equation must be the same. Take, for example, volume (V) which should have units of length cubed (L^3).
A correct formula for the volume of a cylinder would be V = πr²h, since radius squared (r²) provides an area (L²), and when multiplied by height (h), gives a volume (L² · L = L³). On the other hand, V = ν² would not be dimensionally consistent since ² does not have any relevant dimensional meaning in this context.
In scalar quantities like volume, mass, and cross-sectional area, a single number and the appropriate unit completely describe the quantity. However, when physical conditions like temperature change, these scalar quantities can also change. For instance, if the temperature increases, materials typically expand, leading to increases in the volume and cross-sectional area of an object.