How do you use differentials to estimate the maximum error in calculating the surface area of the box if the dimensions of a closed rectangular box are measured as 60 centimeters, 80 centimeters, and 90 centimeters, respectively, with the error in each measurement at most .2 centimeters?

1 Answer
Jim H
Mar 9, 2015

The answer is #184# #cm^2#, but here's how:

The surface area of a closed rectangular box is #A = 2LW+2WH+2LH#,
where #L#=length, #W#=width, #H#=height.

The differential of area is used as the approximate maximum error.
#A = 2[LW+WH+LH]#.

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

Each measurement has same maximum error, all of the differentials are the same. Namely, #0.2# #cm#, so
#dL=dW=dH=0.2# #cm#.
#L=60# #cm# , #W=80# #cm# and #H=90# #cm#.

Substituting into:
#dA=2*[((dL)W+L(dW))+((dW)H+W(dH))+((dL)H+L(dH))]#.
gives:
#dA=2*[((0.2)80+60(0.2))+((0.2)90+80(0.2))+((0.2)90+60(0.2))]#
#=0.4[80+60+90+80+90+60]#
#=0.4(460)=184# #cm^2#.

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