[tex]\large\underline{\sf{Solution-}}[/tex]
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
[tex]\rm\implies \:\dfrac{\triangle d}{d} \times 100 \: = \: 1\% \\ [/tex]
Now, we know area (A) of circle of diameter d is
[tex]\rm \: A \: = \: \dfrac{\pi}{4} {d}^{2} \\ [/tex]
Taking log on both sides, we get
[tex]\rm \: logA \: = \: log\bigg(\dfrac{\pi}{4} {d}^{2}\bigg) \\ [/tex]
[tex]\rm \: logA = log\pi - log4 + log {d}^{2} \\ [/tex]
[tex]\rm \: logA = log\pi - log4 + 2logd \\ [/tex]
On differentiating both sides, we get
[tex]\rm \: \dfrac{\triangle A}{A} = 2\dfrac{\triangle d}{d} \\ [/tex]
can be further rewritten as
[tex]\rm \: \dfrac{\triangle A}{A} \times 100 = 2\dfrac{\triangle d}{d} \times 100 \\ [/tex]
[tex]\rm \: \dfrac{\triangle A}{A} \times 100 = 2 \times 1\% \\ [/tex]
[tex]\rm\implies \:\rm \: \dfrac{\triangle A}{A} \times 100 = 2\% \\ [/tex]
Hence, percentage error in the measurement of its area is 2 %
[tex]\rule{190pt}{2pt}[/tex]
Formulae Used :-
[tex]\boxed{ \rm{ \:logxy = logx + logy \: }} \\ [/tex]
[tex]\boxed{ \rm{ \:log \frac{x}{y} = logx - logy \: }} \\ [/tex]
[tex]\boxed{ \rm{ \:log {x}^{y} = y \: logx \: }} \\ [/tex]
[tex]\boxed{ \rm{ \:\dfrac{d}{dx}logx = \frac{1}{x} \: }} \\ [/tex]
[tex]\boxed{ \rm{ \:\dfrac{d}{dx}k = 0 \: }} \\ [/tex]
Additional Information :-
[tex]\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}[/tex]
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,
[tex]\Longrightarrow\boxed{\bold{Area_{(circular\:disk)} = \pi r^2}}[/tex]
Percentage error in the measurement:
[tex]\Longrightarrow \huge{\frac{\triangle A}{A} }\times 100[/tex]
[tex]\Longrightarrow 2 \times \frac{\triangle r}{r}\times100[/tex]
[tex]\Longrightarrow 2 \times 1\%[/tex]
[tex]\Longrightarrow 2 \%[/tex]
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Verified answer
[tex]\large\underline{\sf{Solution-}}[/tex]
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
[tex]\rm\implies \:\dfrac{\triangle d}{d} \times 100 \: = \: 1\% \\ [/tex]
Now, we know area (A) of circle of diameter d is
[tex]\rm \: A \: = \: \dfrac{\pi}{4} {d}^{2} \\ [/tex]
Taking log on both sides, we get
[tex]\rm \: logA \: = \: log\bigg(\dfrac{\pi}{4} {d}^{2}\bigg) \\ [/tex]
[tex]\rm \: logA = log\pi - log4 + log {d}^{2} \\ [/tex]
[tex]\rm \: logA = log\pi - log4 + 2logd \\ [/tex]
On differentiating both sides, we get
[tex]\rm \: \dfrac{\triangle A}{A} = 2\dfrac{\triangle d}{d} \\ [/tex]
can be further rewritten as
[tex]\rm \: \dfrac{\triangle A}{A} \times 100 = 2\dfrac{\triangle d}{d} \times 100 \\ [/tex]
[tex]\rm \: \dfrac{\triangle A}{A} \times 100 = 2 \times 1\% \\ [/tex]
[tex]\rm\implies \:\rm \: \dfrac{\triangle A}{A} \times 100 = 2\% \\ [/tex]
Hence, percentage error in the measurement of its area is 2 %
[tex]\rule{190pt}{2pt}[/tex]
Formulae Used :-
[tex]\boxed{ \rm{ \:logxy = logx + logy \: }} \\ [/tex]
[tex]\boxed{ \rm{ \:log \frac{x}{y} = logx - logy \: }} \\ [/tex]
[tex]\boxed{ \rm{ \:log {x}^{y} = y \: logx \: }} \\ [/tex]
[tex]\boxed{ \rm{ \:\dfrac{d}{dx}logx = \frac{1}{x} \: }} \\ [/tex]
[tex]\boxed{ \rm{ \:\dfrac{d}{dx}k = 0 \: }} \\ [/tex]
[tex]\rule{190pt}{2pt}[/tex]
Additional Information :-
[tex]\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}[/tex]
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,
[tex]\Longrightarrow\boxed{\bold{Area_{(circular\:disk)} = \pi r^2}}[/tex]
Percentage error in the measurement:
[tex]\Longrightarrow \huge{\frac{\triangle A}{A} }\times 100[/tex]
[tex]\Longrightarrow 2 \times \frac{\triangle r}{r}\times100[/tex]
[tex]\Longrightarrow 2 \times 1\%[/tex]
[tex]\Longrightarrow 2 \%[/tex]