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Conical Tank Draining

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About This MicroSim

This visualization demonstrates one of the most classic related rates problems in calculus: how fast does the water level drop in a conical tank? The key insight is using similar triangles to eliminate a variable before differentiating.

The Problem Setup

A conical tank (vertex down) has:

  • H = tank height (total)
  • R = tank radius at the top
  • Water draining at rate dV/dt (negative because volume decreases)

We want to find dh/dt - how fast the water level is changing.

The Similar Triangles Key

The cross-section of a cone creates similar triangles. At any moment, the water forms a smaller cone inside the tank. The ratio of dimensions is constant:

\[\frac{r}{h} = \frac{R}{H}\]

This lets us express the water radius in terms of height:

\[r = \frac{R}{H} \cdot h\]

Delta Moment

"See those two triangles? They're like copies at different scales! That's why the ratio r/h always equals R/H. This little relationship is our ticket to solving the whole problem!"

Deriving the Formula

Step 1: Write volume in terms of h only

\[V = \frac{1}{3}\pi r^2 h = \frac{1}{3}\pi \left(\frac{R}{H}h\right)^2 h = \frac{\pi R^2}{3H^2}h^3\]

Step 2: Differentiate with respect to time

\[\frac{dV}{dt} = \frac{\pi R^2}{H^2}h^2 \cdot \frac{dh}{dt}\]

Step 3: Solve for dh/dt

\[\frac{dh}{dt} = \frac{H^2}{\pi R^2 h^2} \cdot \frac{dV}{dt}\]

Key Insight

Notice that dh/dt depends on h! When the tank is nearly full (h is large), the water level drops slowly. When almost empty (h is small), the level drops much faster - even though dV/dt is constant!

Delta's Sidequest

"Wait, the water level speeds up as it drains? But the drain rate is constant! Oh, I see - when h is small, there's less surface area, so the same volume loss means more height change. Clever, physics!"

How to Use

  1. H Slider: Adjust the total tank height (5-15 ft)
  2. R Slider: Adjust the tank radius at top (2-6 ft)
  3. dV/dt Slider: Set the drainage rate (-5 to -0.5 ft^3/min)
  4. Play/Pause: Start or stop the draining animation
  5. Reset: Return to default values
  6. Click Tank: Click anywhere on the tank to set the water level

What to Observe

  • The blue region shows current water level
  • Labels display all dimensions: H, R, h, r
  • Similar triangles diagram highlights the key relationship
  • Formulas panel shows the mathematical derivation
  • Values panel shows real-time calculations including dh/dt
  • Watch how dh/dt accelerates as water level drops!

Lesson Plan

Learning Objectives

After using this MicroSim, students will be able to:

  1. Apply similar triangles to eliminate variables in related rates problems (Bloom Level 3)
  2. Calculate how fast the water level changes given tank dimensions and drain rate
  3. Understand why the rate of water level change depends on current water height

Prerequisite Knowledge

  • Volume formula for a cone: \(V = \frac{1}{3}\pi r^2 h\)
  • Similar triangles and proportional relationships
  • Chain rule for differentiation
  • Basic related rates setup

Suggested Activities

  1. Predict First: Before running the simulation, have students predict: Will the water level drop faster when the tank is full or nearly empty? Then verify with the simulation.

  2. Compare Tank Shapes: Try different H and R combinations. Which tank shape causes the fastest level drop? Why?

  3. Calculate by Hand: Pause at a specific water height. Calculate dh/dt using the formula, then verify against the displayed value.

  4. Real-World Connection: Research actual conical tanks (traffic cones filled with water, funnels, rocket fuel tanks). Why might engineers care about this rate?

Discussion Questions

  1. Why must we eliminate r before differentiating?
  2. What happens to dh/dt as h approaches 0? Is this physically realistic?
  3. How would this problem change if the cone were vertex-up (like an ice cream cone)?
  4. If you wanted the water level to drop at a constant rate, what would need to change?

Assessment Questions

  1. A conical tank has H = 12 ft and R = 4 ft. Water drains at 3 ft^3/min. Find dh/dt when h = 6 ft.

  2. Explain why the similar triangles relationship r/h = R/H is essential to solving this problem.

  3. If the tank radius is doubled while keeping height the same, how does this affect dh/dt?

Embedding

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<iframe src="https://dmccreary.github.io/calculus/sims/conical-tank-draining/main.html"
        height="570px" width="100%" scrolling="no"></iframe>

References