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The error in the measurement of diameter of a circular disk is 1%. What is the percentage error in the measurement of its area?

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Answer 1

$\large\underline{\sf{Solution-}}$

Let assume that diameter of circular disc be d units.

It is given that, error in the measurement of diameter of a circular disk is 1%.

It means

$\rm\implies \:\dfrac{\triangle d}{d} \times 100 \: = \: 1\%$

Now, we know area (A) of circle of diameter d is

$\rm \: A \: = \: \dfrac{\pi}{4} {d}^{2}$

Taking log on both sides, we get

$\rm \: logA \: = \: log\bigg(\dfrac{\pi}{4} {d}^{2}\bigg)$

$\rm \: logA = log\pi - log4 + log {d}^{2}$

$\rm \: logA = log\pi - log4 + 2logd$

On differentiating both sides, we get

$\rm \: \dfrac{\triangle A}{A} = 2\dfrac{\triangle d}{d}$

can be further rewritten as

$\rm \: \dfrac{\triangle A}{A} \times 100 = 2\dfrac{\triangle d}{d} \times 100$

$\rm \: \dfrac{\triangle A}{A} \times 100 = 2 \times 1\%$

$\rm\implies \:\rm \: \dfrac{\triangle A}{A} \times 100 = 2\%$

Hence, percentage error in the measurement of its area is 2 %

$\rule{190pt}{2pt}$

Formulae Used :-

$\boxed{ \rm{ \:logxy = logx + logy \: }}$

$\boxed{ \rm{ \:log \frac{x}{y} = logx - logy \: }}$

$\boxed{ \rm{ \:log {x}^{y} = y \: logx \: }}$

$\boxed{ \rm{ \:\dfrac{d}{dx}logx = \frac{1}{x} \: }}$

$\boxed{ \rm{ \:\dfrac{d}{dx}k = 0 \: }}$

$\rule{190pt}{2pt}$

Additional Information :-

$\begin{gathered}\: \: \: \: \: \: \begin{gathered}\begin{gathered} \footnotesize{\boxed{ \begin{array}{cc} \small\underline{\frak{\pmb{ \red{More \: Formulae}}}} \\ \\ \bigstar \: \bf{ log_{x}(x) = 1}\\ \\ \bigstar \: \bf{ log_{x}( {x}^{y} ) = y}\\ \\ \bigstar \: \bf{ log_{ {x}^{z} }( {x}^{w} ) = \dfrac{w}{z} }\\ \\ \bigstar \: \bf{ log_{a}(b) = \dfrac{logb}{loga} }\\ \\ \bigstar \: \bf{ {e}^{logx} = x}\\ \\ \bigstar \: \bf{ {e}^{ylogx} = {x}^{y}}\\ \\ \bigstar \: \bf{log1 = 0}\\ \\ \end{array} }}\end{gathered}\end{gathered}\end{gathered}$

Answer 2

Answer:

2%

Step-by-step explanation:

Given:-

The error in the measurement:-

Diameter of a circular disk = 1%

Then,

Radius of a circular disk = 1/2

To Find:-

What is the percentage error in the measurement of its area?

Solution:-

We know,

$\Longrightarrow\boxed{\bold{Area_{(circular\:disk)} = \pi r^2}}$

Percentage error in the measurement:

$\Longrightarrow \huge{\frac{\triangle A}{A} }\times 100$

$\Longrightarrow 2 \times \frac{\triangle r}{r}\times100$

$\Longrightarrow 2 \times 1\%$

$\Longrightarrow 2 \%$