SATHEE: Differential Equations 3 Question 10

Differential Equations 3 Question 10

10. A right circular cone with radius $R$ and height $H$ contains a liquid which evaporates at a rate proportional to its surface area in contact with air (proportionality constant $= k > 0$ ). Find the time after which the cone is empty.

$(2003, 4$ M)

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Answer:

Correct Answer: 10. $T = \frac{H}{k}$

Solution:

Given, liquid evaporates at a rate proportional to its surface area.

$\Rightarrow \frac{d V}{d t} \propto - S$

We know that, volume of cone $= \frac{1}{3} π r^{2} h$

and surface area $= π r^{2}$

or $V = \frac{1}{3} π r^{2} h$ and $S = π r^{2}$

Where,

$\tan θ = \frac{R}{H} and \frac{r}{h} = \tan θ$

From Eqs. (ii) and (iii), we get

$V = \frac{1}{3} π r^{3} \cot θ and S = π r^{2}$

On substituting Eq. (iv) in Eq. (i), we get

$\begin{aligned} \frac{1}{3} \cot θ \cdot 3 r^{2} \frac{d r}{d t} & = - k π r^{2} \\ \Rightarrow & \cot θ \int_{R}^{0} d r & = - k \int_{0}^{T} d t \\ \Rightarrow & \cot θ (0 - R) & = - k (T - 0) \\ \Rightarrow & R \cot θ & = k T \Rightarrow H = k T [from Eq. (iii)] \\ \Rightarrow & T & = \frac{H}{k} \end{aligned}$

$∴$ Required time after which the cone is empty, $T = \frac{H}{k}$

Differential Equations 3 Question 10

10. A right circular cone with radius $R$ and height $H$ contains a liquid which evaporates at a rate proportional to its surface area in contact with air (proportionality constant $= k > 0$ ). Find the time after which the cone is empty.

Answer:

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Differential Equations 3 Question 10

10. A right circular cone with radius R and height H contains a liquid which evaporates at a rate proportional to its surface area in contact with air (proportionality constant =k>0 ). Find the time after which the cone is empty.

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

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10. A right circular cone with radius $R$ and height $H$ contains a liquid which evaporates at a rate proportional to its surface area in contact with air (proportionality constant $= k > 0$ ). Find the time after which the cone is empty.