Applications of the Derivative - Theory & Concepts - Cone Related Rates

Apply differentiation to solve problems in motion, optimization, geometric analysis, and marginal cost.

A standard civil engineering application: analyze how depth $h$ changes over time as water flows into a conical tank at a constant rate $dV/dt$ .

Tank Total Height (

H

): 10 m

Tank Top Radius (

R

): 5 m

Inflow Rate (

Q = dV/dt

): 1.5 m³/min

Current Water Depth (

h

): 4.00 m

0.1m10m (Full)

Governing Equation

\frac{dh}{dt} = \frac{dV/dt}{\pi r^2}

Current Radius $r$:

2.00 m

Water Volume $V$:

16.8 m³

Rate dh/dt:0.1194 m/min

Reservoir Geometry Cross-Section

Depth vs. Depth Rate (dh/dt)

Loading chart...

Key Insight: Observe how the green curve decreases exponentially. As the depth

h

increases, the cross-sectional area expands, meaning the rate of vertical rise

dh/dt

slows down drastically for the same constant inflow!

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