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Question Number 190249 by alcohol last updated on 30/Mar/23
if 8% error is made on x,   what is the percentage error  on λx^(1/5)  ?
$${if}\:\mathrm{8\%}\:{error}\:{is}\:{made}\:{on}\:{x},\: \\ $$$${what}\:{is}\:{the}\:{percentage}\:{error} \\ $$$${on}\:\lambda{x}^{\frac{\mathrm{1}}{\mathrm{5}}} \:? \\ $$
Answered by mr W last updated on 30/Mar/23
y=λx^(1/5)   ((Δy)/(Δx))≈(dy/dx)=(λ/5)x^(−(4/5))   ((Δy)/y)=(((λ/5)x^(−(4/5)) )/(λx^(1/5) ))=((Δx)/(5x))=((8%)/5)=1.6%  i.e. error on λx^(1/5)  is 1.6%.
$${y}=\lambda{x}^{\frac{\mathrm{1}}{\mathrm{5}}} \\ $$$$\frac{\Delta{y}}{\Delta{x}}\approx\frac{{dy}}{{dx}}=\frac{\lambda}{\mathrm{5}}{x}^{−\frac{\mathrm{4}}{\mathrm{5}}} \\ $$$$\frac{\Delta{y}}{{y}}=\frac{\frac{\lambda}{\mathrm{5}}{x}^{−\frac{\mathrm{4}}{\mathrm{5}}} }{\lambda{x}^{\frac{\mathrm{1}}{\mathrm{5}}} }=\frac{\Delta{x}}{\mathrm{5}{x}}=\frac{\mathrm{8\%}}{\mathrm{5}}=\mathrm{1}.\mathrm{6\%} \\ $$$${i}.{e}.\:{error}\:{on}\:\lambda{x}^{\frac{\mathrm{1}}{\mathrm{5}}} \:{is}\:\mathrm{1}.\mathrm{6\%}. \\ $$
Commented by mr W last updated on 30/Mar/23
or  y=λx^(1/5)   y+Δy=λ(x+Δx)^(1/5) ≈λx^(1/5) (1+((Δx)/(5x)))=y(1+((Δx)/(5x)))  Δy≈y((Δx)/(5x))  ((Δy)/y)≈((Δx)/(5x))=((8%)/5)=1.6%
$${or} \\ $$$${y}=\lambda{x}^{\frac{\mathrm{1}}{\mathrm{5}}} \\ $$$${y}+\Delta{y}=\lambda\left({x}+\Delta{x}\right)^{\frac{\mathrm{1}}{\mathrm{5}}} \approx\lambda{x}^{\frac{\mathrm{1}}{\mathrm{5}}} \left(\mathrm{1}+\frac{\Delta{x}}{\mathrm{5}{x}}\right)={y}\left(\mathrm{1}+\frac{\Delta{x}}{\mathrm{5}{x}}\right) \\ $$$$\Delta{y}\approx{y}\frac{\Delta{x}}{\mathrm{5}{x}} \\ $$$$\frac{\Delta{y}}{{y}}\approx\frac{\Delta{x}}{\mathrm{5}{x}}=\frac{\mathrm{8\%}}{\mathrm{5}}=\mathrm{1}.\mathrm{6\%} \\ $$

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