Name: Local Maxima and Minima of a Function
Uploaded: 2019-07-19
Duration: 14 min 17 s
Description: Do you know what is minima or maxima of a function? This video also explains how to find local maxima and minima by differentiation in a detailed manner.

Question

A conical vessel whose height is

4

meters and of base radius

2

meters is being filled with water at the rate of

0.75

cubic meters per minute. Find the rate at which the level of the water is rising when the depth of water is

1.5

meters.

Accepted Answer

Let $OA$ be height of conical vessel, $AB$ be radius of conical vessel.

$v$ be volume of water cone. $r$ be radius of water cone and $h$ be height of water cone.

Given $OA = 4 m, AB = 2 m, \frac{dv}{dt} = 0.75 m^{3} / \min$

Now, $v = \frac{1}{3} {πr}^{2} h . . . . . . . . . . . . . (1)$

From figure, In $∆ OAB$ and $OA' B'$

$∠ O = ∠ O$ [common angle]

$∠ OAB = ∠ OA' B'$ [both are $90 °$ ]

So, by $AA$ similarity, criterion

$∆ OAB ~ ∆ OA' B'$

$\Rightarrow \frac{OA}{OA'} = \frac{AB}{A' B'}$ [corresponding sides of similar triangles are proportional]

$\Rightarrow \frac{4}{h} = \frac{2}{r}$

$\Rightarrow r = \frac{h}{2}$ .

Put $r = \frac{h}{2}$ in equation $(1)$

$\Rightarrow v = \frac{1}{3} π \frac{h^{2}}{4} \cdot h = \frac{π}{3} \cdot \frac{h^{3}}{4}$

Differentiate both side with respect to $t$

$\frac{dv}{dt} = \frac{π}{12} \cdot 3 h^{2} \cdot \frac{dh}{dt}$

Since, $\frac{dv}{dt} = 0.75 m^{3} / \min$ and when $h = 1.5 m$

$0.75 = \frac{π}{12} \cdot 3 \cdot {(\frac{3}{2})}^{2} \cdot \frac{dh}{dt}$

$\frac{3}{4} = \frac{π}{4} \cdot \frac{9}{4} \cdot \frac{dh}{dt}$ $\Rightarrow \frac{dh}{dt} = \frac{4}{3 π} m / \min$ .

A conical vessel whose height is $4$ meters and of base radius $2$ meters is being filled with water at the rate of $0.75$ cubic meters per minute. Find the rate at which the level of the water is rising when the depth of water is $1.5$ meters.

Important Questions on Applications of Derivatives

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