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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. 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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