What is the volume of the water in terms of…

What is the volume of the water in terms of its depth h?

Question 114

EasyOriginal question

Water is poured at a steady 2 m³/min into a tank shaped like a cone, held vertex down. The tank is 10 m deep and the radius of its top is 5 m. When the water is $h$ m deep, its surface is a circle of radius $r$ m.

What is the volume of the water in terms of its depth $h$ ?

\frac{\pi h^{3}}{3}

Correct answerYour answer

\frac{4\pi h^{3}}{3}

Correct answerYour answer

\frac{\pi h^{3}}{6}

Correct answerYour answer

\frac{\pi h^{3}}{12}

Correct answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer D

Correct answer: D

Similar triangles: $\frac{r}{h} = \frac{5}{10}$ , so $r = \frac{h}{2}$ .
$V = \frac{1}{3}\pi r^{2}h = \frac{1}{3}\pi \cdot \frac{h^{2}}{4} \cdot h = \frac{\pi h^{3}}{12}$ .

Worth noting

$\frac{\pi h^{3}}{3}$ takes $r = h$ ; the tank's shape gives $r = \frac{h}{2}$ .

Concept: Writing a volume in one variable
Typical timeTypical: 90 s
Your timeYours: —

The rule this testsRate of change of quantities

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Ten questions from Chain Rule, Log Differentiation & Rate of Change, at your own pace.

What is the volume of the water in terms of its depth h?

Question 114

Explanation