Question

A student is trying to remember some formulas from geometry. In what follows, assume A is area, V is volume, and all other variables are lengths. Determine which formulas are dimensionally consistent.

(a) V=πr2h;
(b) A=2πr2+2πrh;
(c) V=0.5bh;
(d) V=πd2;
(e) V=πd3/6.

Answer 1

To determine which formulas are dimensionally consistent, we need to check if the units of measurement on both sides of the equation match. The dimensionally consistent formulas are (a) V = πr²h and (b) A = 2πr² + 2πrh.

Step-by-step explanation:

To determine which formulas are dimensionally consistent, we need to check if the units of measurement on both sides of the equation match. Let's go through each formula:

1. V = πr²h: The units of volume are length cubed (L^3), while the units of radius squared times height are length squared times length (L^2 × L = L^3). So this formula is dimensionally consistent.
2. A = 2πr² + 2πrh: The units of area are length squared (L^2), while the units of the first term are length squared and the units of the second term are length times length (L^2 + L^2 = L^2). So this formula is dimensionally consistent.
3. V = 0.5bh: The units of volume are length cubed (L^3), while the units of base times height are length times length (L × L = L^2). So this formula is not dimensionally consistent.
4. V = πd²: The units of volume are length cubed (L^3), while the units of diameter squared are length squared (L^2). So this formula is not dimensionally consistent.
5. V = πd^3/6: The units of volume are length cubed (L^3), while the units of diameter cubed divided by 6 are length cubed divided by 6 (L^3/6). So this formula is dimensionally consistent.

Therefore, the dimensionally consistent formulas are (a) V = πr²h and (b) A = 2πr² + 2πrh.