Final answer:
To solve for height h in the equation v=\u03c0r^2h, divide both sides by \u03c0r^2 to get h = v/(\u03c0r^2), ensuring dimensions are consistent as the volume and height are dimensionally related.
Step-by-step explanation:
To solve the equation v=\u03c0r^2h for height h as a function of the radius r and volume v, you need to isolate h on one side of the equation. This can be done by dividing both sides by \u03c0r^2. Thus, the formula for h will be h = \frac{v}{\pi r^2}.
Dimensional analysis can be applied in this context to confirm the consistency of the formula. A volume, v, has dimensions of L^3 as it represents the three-dimensional space an object occupies. The radius, r, has a dimension of L (length), and when this is squared as in r^2, it has dimensions of L^2. Multiplying this area-like term (L^2) by another length term, h, would give L^3, thus confirming that the original equation is dimensionally consistent.