Question

guys please and this two questions as soon as possible​

Answer 1

$\underline{\rm\red{Question :}}$

If the percentage error in the measurement of side of cube is 2% , then the percentage error in its volume and area are -

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$\underline{\rm\red{Answer - }}$

$\underline{\rm\pink{Given}}$ -

$\implies$Percentage error in side = $\rm \dfrac{\Delta s}{s} = 2\%$

$\underline{\rm\pink{To\: find - }}$

Percentage error in volume and area

$\underline{\rm\pink{Solution -}}$

$\rm Volume = s^3$

Percentage error in volume = 3 ( Percentage error in each side )

$\implies$$\rm \dfrac{\Delta v}{v} = 3 \dfrac{\Delta s}{s}$

$\implies$$\rm \dfrac{\Delta v}{v} = 3 \times 2\%$

$\implies$$\rm \dfrac{\Delta v}{v} = 6\%$

$\underline{\underline{\sf\purple{Percentage \:error\: in \:Volume = 6\%}}}$

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$\rm Area = s^2$

Percentage error in area = 2 ( Percentage error in each side )

$\implies$$\rm \dfrac{\Delta a}{a} = 2 \dfrac{\Delta s}{s}$

$\implies$$\rm \dfrac{\Delta a}{a} = 2 \times 2\%$

$\implies$$\rm \dfrac{\Delta a}{a} = 4\%$

$\underline{\underline{\sf\purple{Percentage \:error \:in \:Area = 4\%}}}$

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$\underline{\rm\red{Question - }}$

The distance covered by particles varies with the time as $\rm x = at^2 - bt^3$ , the time of body when acceleration is zero is

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$\underline{\rm\red{Answer - }}$

$\underline{\rm\pink{Given - }}$

$\rm x = at^2 - bt^3$

$\underline{\rm\pink{To\: find - }}$

Time at acceleration = 0

$\underline{\rm\pink{Solution -}}$

Acceleration is given by -

$\rm a = \dfrac{d^2x}{dt^2}$

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By double differnetiation of equation and then equating it to 0 will give the value of time -

$\implies$$\rm \dfrac{dx}{dt} = \dfrac{d( {at}^{2} - {bt}^{3}) }{dt}$

$\implies$$\rm \dfrac{dx}{dt} = 2at - 3b {t}^{2}$

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$\implies$$\rm \dfrac{d^2x}{dt^2} = \dfrac{d( 2at - 3bt^2) }{dt}$

$\implies$$\rm \dfrac{d^2x}{dt^2} = 2a - 6bt$

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$\implies$$\rm a = \dfrac{d^2x}{dt^2} = 2a - 6bt$

$\implies$$\rm 2a - 6bt = 0$

$\implies$$\rm 6bt = 2a$

$\implies$$\rm 3bt = a$

$\implies$$\rm t = \dfrac{a}{3b}$

$\underline{\underline{\sf\purple{Time \:when \:acceleration\: is \:zero = \dfrac{a}{3b}}}}$

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