Final answer:
The correct formula to find the lateral area of a right cone is LA = πrs, where r is the radius and s is the slant height. The units must be dimensionally consistent, meaning areas are in square units and volumes are in cubic units. For spheres, volume is given by 4/3 (π) (r)^3 and surface area by 4 (π) (r)^2.
Step-by-step explanation:
The correct formula for finding the lateral area (LA) of a right cone, given the radius (r) and the slant height (s), is LA = πrs. This is found by unwrapping the lateral surface of a cone, which results in a sector of a circle, and the area of that sector is proportional to the circumference of the base of the cone multiplied by the slant height. Thus, we use πr for the circumference of the base and multiply it by s to find the lateral area, resulting in the formula LA = πrs.
It's important to understand that when we talk about dimensional consistency in geometry, it means that the units on both sides of the equation must match. For example, for area, we would expect the units to be squared units (like meters squared in the case of a 2-dimensional surface). For the given formulas to be dimensionally consistent, option (C) V = 0.5bh correctly represents the volume of a triangle-based pyramid.
In terms of the cross-sectional area related to resistance of a cylinder, the formula R = ΠL can be rearranged to find the cross-sectional area (A). Where resistance (R), resistivity (Π), and length (L) are considered, we arrive at the area equation A = ΠL/R.
Finally, when differentiating between the formulas for a sphere, volume = 4/3 (π) (r)^3 is dimensionally consistent for volume, for it represents cubic units, while surface area = 4 (π) (r)^2 represents squared units and is therefore the formula for surface area.