Related rates, question 19

Topic: Related rates
Difficulty: Challenging
Suggested time: 6 min
Marks: 4
Theory: UBC Math 100 §5 CLP-1 §3.2Key results

Water drains from an inverted right circular cone at 8 cm³/s. The cone is 12 cm tall and 6 cm in radius at the top. At what rate is the water level changing when the water is 4 cm deep?

V = \frac{1}{3}\pi r^{2} h

Your answer

cm/s

A decimal to two places is enough, negative if the level is falling.

How sure are you?

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Theory UBC Math 100 §5 CLP-1 §3.2Key results

Worked solution

Strategy

Reduce the volume to a function of a single variable using the fixed shape of the cone, then differentiate with respect to time and solve for the rate of the water level.

Working

$\frac{r}{h} = \frac{6}{12} = \frac{1}{2} \implies r = \frac{h}{2}$
The water forms a cone similar to the container, so the ratio of radius to depth is fixed.
$V = \frac{1}{3}\pi\left(\frac{h}{2}\right)^{2}h = \frac{\pi h^{3}}{12}$
Substitute to get the volume in terms of depth alone.
$\frac{dV}{dt} = \frac{\pi h^{2}}{4}\cdot\frac{dh}{dt}$
Differentiate with respect to time.
$\frac{dh}{dt} = \frac{4}{\pi h^{2}}\cdot\frac{dV}{dt} = \frac{4(-8)}{\pi (4)^{2}}$
Draining means dV/dt is negative. Substitute h = 4.
$\frac{dh}{dt} = -\frac{32}{16\pi} = -\frac{2}{\pi} \approx -0.637 \ \text{cm/s}$
The level is falling at about 0.64 cm/s at this depth.

Answer

\frac{dh}{dt} = -\frac{2}{\pi} \approx -0.637 \ \text{cm/s}

Check

At 4 cm deep the water surface is a disc of radius 2 cm, area 4π ≈ 12.57 cm², and losing 8 cm³/s through it lowers the level about 8/(4π) ≈ 0.637 cm/s.

Where this goes wrong

Treating r as a constant 6 and differentiating V = 12πh. The radius of the water surface changes as the level drops, so r must be written in terms of h before differentiating.

Leans on

Similar triangles, from Functions, graphs and inverses

Try a similar question

A water trough is 6 m long, and its cross-section is an isosceles trapezoid 1 m wide at the bottom, 2 m wide at the top, and 1 m deep. Water is pumped in at 0.3 m³/min. How fast is the water level rising when the water is 0.5 m deep?

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Related rates practice question 19