To determine how long it takes to lower the liquid level from $h = 5 \, \text{m}$ to $h = 2 \, \text{m}$ in a truncated cone, we need to approach the problem using the concept of fluid flow, typically modeled by Torricelli's law. First, let's establish some basic geometric and physical relationships.

Assumptions and variables:

• $h$ is the height of the liquid at any time.
• $r_1$ is the radius of the surface at $h = 5 \, \text{m}$.
• $r_2$ is the radius of the surface at $h = 2 \, \text{m}$.
• The cone has a half-angle $\theta = 60^\circ$.
• The flow rate of the liquid follows Torricelli’s law: $v = \sqrt{2g h}$ where $v$ is the velocity of the liquid exiting the tank, $g$ is the acceleration due to gravity ($g = 9.81 \, \text{m/s}^2$), and $h$ is the height of the liquid.

Step 1: Relate the radius to the height

Given that the cone has a half-angle $\theta = 60^\circ$, we can use the geometry of the cone to relate the radius at any height $h$: $r(h) = h \tan(60^\circ) = h \sqrt{3}$ So, the radius of the liquid surface at any height $h$ is $r(h) = h \sqrt{3}$.

Step 2: Volume of liquid in the cone

The volume $V$ of liquid at any height $h$ is the volume of a truncated cone: $V = \frac{1}{3} \pi h (r(h))^2 = \frac{1}{3} \pi h \left( h \sqrt{3} \right)^2 = \pi \sqrt{3} h^3$

Step 3: Apply Torricelli’s law and continuity equation

According to Torricelli’s law, the velocity of the liquid exiting the hole is proportional to the square root of the height: $v = \sqrt{2gh}$ The flow rate $Q$ is given by the area of the hole times the velocity: $Q = A_{\text{hole}} \sqrt{2gh}$ where $A_{\text{hole}}$ is the area of the hole through which the liquid is exiting.

Step 4: Time to drain from 5 m to 2 m

The rate of change of the liquid volume is related to the flow rate: $\frac{dV}{dt} = -Q = -A_{\text{hole}} \sqrt{2gh}$ Since $V = \pi \sqrt{3} h^3$, we can differentiate this with respect to $t$: $\frac{dV}{dh} \cdot \frac{dh}{dt} = -A_{\text{hole}} \sqrt{2gh}$ This simplifies to an equation involving $h$ and $t$. Solving this differential equation gives the time required to drain the liquid from 5 m to 2 m.

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5 Related Questions:

1. What is Torricelli's law, and how does it apply to draining fluids?
2. How is the volume of a truncated cone calculated for any given height?
3. What factors influence the time required for fluid to drain from a vessel?
4. How does the cone's angle affect the radius-height relationship?
5. What happens if the hole size or gravitational force changes in the draining process?

Tip: Understanding how geometric shapes affect fluid flow helps in many real-world applications, like designing tanks or reservoirs.