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 %