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If the density of cube is found by measuring its mass and length of its side. If maximum errors in the measurement of mass and length are 0.1% and 0.3% respectively, then find out the maximum error in measurement of density.
Density D = mass/Volume or, D = m/l3 Take log of both sides. log(D) = log(m) - 3 log(l) Now, differentiate both sides. Differential of log(x) is 1/x dx. x can be anything. So, here, we have (1/D) dD =(1/m) dm - 3 (1/l) dl [Because log(1/2) is a constant, and derivative of any constant is zero] Now, dK is nothing but delta(K). So, dK is the error in measurement of K. dK/K is the relative error in measurement of K. Percentage error is relative error x 100 Hence, dK/K = dm/m + 3 dl/l [Note that the minus sign changes to plus, because errors always add up, unlike derivatives] We are given that dm/m = 0.1% = 0.001 Also, dl/l = 0.3% = 0.003 Therefore, Relative error in measurement of K = dK/K = 0.001 + 3(0.003) = 0.001 + 0.009 = 0.01 Percentage error = 0.01 x 100 = 1 %
Density D = mass/Volume
or, D = m/l3 Take log of both sides. log(D) = log(m) - 3 log(l) Now, differentiate both sides. Differential of log(x) is 1/x dx. x can be anything. So, here, we have (1/D) dD =(1/m) dm - 3 (1/l) dl [Because log(1/2) is a constant, and derivative of any constant is zero] Now, dK is nothing but delta(K). So, dK is the error in measurement of K. dK/K is the relative error in measurement of K. Percentage error is relative error x 100 Hence, dK/K = dm/m + 3 dl/l [Note that the minus sign changes to plus, because errors always add up, unlike derivatives] We are given that dm/m = 0.1% = 0.001 Also, dl/l = 0.3% = 0.003
Therefore, Relative error in measurement of K = dK/K = 0.001 + 3(0.003) = 0.001 + 0.009 = 0.01 Percentage error = 0.01 x 100 = 1 %
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