Dimensional analysis confirms that equations (a), (b), (c), and (e) are dimensionally consistent, as they correctly relate volume or area with products of lengths to the appropriate powers. Equation (d) is not consistent due to a mismatch in dimensions.
Considering the formulas provided, we need to determine which are dimensionally consistent. Dimensional analysis helps verify that equations make sense by checking that each term on both sides of an equation has the same dimensions.
(a) V = πr²h; This equation represents the volume of a cylinder, where 'V' is volume, 'π' is Pi, 'r' is the radius, and 'h' is the height. Since both sides have the dimension of length³, this equation is dimensionally consistent.(b) A = 2πr² + 2πrh; This equation stands for the surface area of a cylinder, combining the areas of its two circular bases and its lateral surface. As the units on both sides represent an area, or length², it is also dimensionally consistent.(c) V = 0.5bh; This formula calculates the volume of a triangular prism or pyramid, where 'b' is the base length, and 'h' is the height. Since both terms are lengths, the right side should have length³, confirming dimensional consistency.(d) V = ñd²; This expression is dimensionally inconsistent since 'V' is volume (length³), while the right side, being the product of a number and length² (area), does not yield length³.(e) V = λd³ 16; Assuming λ is a constant with no unit dimension, this expression gives the correct dimension of volume (length³) for 'V', making it dimensionally consistent.