The dimensions of a closed rectangular box are measured as 50 centimeters, 90 centimeters, and 80 centimeters, respectively, with the error in each measurement at most. 2 centimeters. Use differentials to estimate the maximum error in calculating the surface area of the box.

Need to get the total surface area of the box

L=length, W=Width, H=height

A = 2LW+2WH+2LH

Take derivative of that

dA=[2 * (dL) * W+2*L * (dW) ]+[2 * (dW) * H+2*W * (dH) ]+[2 * (dL) * H+2*L * (dH) ]

dA is your maximum error.

Since each side has same error its 0.2cm so therefore

dL=dW=dH=0.2cm

L=90cm

W=80cm

H=50cm

Just sub in the value in the above equation, and your resulting value will be your maximum error.

dA=[2 * (0.2) * 80+2*90 * (0.2) ]+[2 * (0.2) * 50+2*80 * (0.2) ]+[2 * (0.2) * 50+2*90 * (0.2) ]

dA=176