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, 100 centimeters, and 90 centimeters, respectively, with the error in each measurement at most .2 centimeters?

1 Answer
Mar 9, 2015

Let's call the measurements x,y,z for now.

Then the total surface area will be
A_0=2*xy+2*xz+2*yz=2(xy+xz+yz)

If we include the differences and set these to d
A_1=2((x+d)(y+d)+(x+d)(z+d)+(y+d)(z+d))

=2((xy+xd+yd+d^2)+(xz+xd+zd+d^2)+(yz+yd+zd+d^2))

Subtract the original surface area A_0

DeltaA=2(xd+yd+d^2+xd+zd+d^2+yd+zd+d^2)
=2(2xd+2yd+2zd+3d^2)

Since d^2 is very small compared to the rest, we can ignore it.
DeltaA=4d(x+y+z) now fill in the numbers:
DeltaA=4*0.2(60+100+90)=0.8*250=200cm^2
For maximum error -0.2 the answer would be -200cm^2

Answer : the maximum error in surface area is 200cm^2
(On a calulated area of 40800cm^2, less than 0.5%)

Remark : if we had taken the d^2 into account the error would be 0.24cm^2 greater (for positive error) or 0.24cm^2 smaller (for negative error).
This is way below the significance range.

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