As a production technician at an industrial chemical plant, you are tasked with setting up an automated sensor system for a new conical storage tank. The system uses the total volume ($V$) of the liquid and the fixed base radius ($r$) of the tank to continuously calculate the liquid's height ($h$). The standard formula for the volume of a cone is $V = \frac{1}{3}\pi r^2 h$. To program the sensor properly, you need to isolate the height variable. Which of the following formulas correctly shows the volume equation rearranged to solve for $h$?

As a construction site assistant, you are tracking the volume of conical gravel piles for a road project. You need to rearrange the standard volume formula, $V = \frac{1}{3}\pi r^2 h$, to solve for the height ($h$) so you can verify pile heights based on their volume and radius. Arrange the following algebraic steps in the correct order to isolate $h$.

A landscape maintenance supervisor needs to calculate the height of conical mulch piles to verify volume deliveries. Starting with the standard volume formula $V = \frac{1}{3}\pi r^2 h$, match each algebraic operation or result with its correct description in the process of solving for the height ($h$).

A warehouse supervisor is using the volume formula $V = \frac{1}{3}\pi r^2 h$ to determine the height ($h$) of a conical grain pile. The supervisor correctly rearranges the formula to isolate the height as $h = \frac{3V}{\pi r^2}$.

Rearranging the Cone Formula for Salt Inventory

An operations supervisor at a bulk landscape supply facility needs to track the inventory of conical compost piles. The standard formula for the volume of a cone is $V = \frac{1}{3}\pi r^2 h$, where $V$ is the volume, $r$ is the radius of the circular base, and $h$ is the height. To easily calculate the height of a pile based on its measured volume and base radius, the supervisor rearranges the formula to isolate $h$:

$h = \frac{\text{Numerator}}{\pi r^2}$

What expression replaces 'Numerator' in this rearranged formula? ____

Rearranging the Cone Volume Formula for Height