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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.

Grade:11

1 Answers

AskIITians Expert Hari Shankar IITD
17 Points
13 years ago

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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