Question

The dimensions of a closed rectangular box are measured as 97 cm, 67 cm, and 34 cm, respectively, with a possible error of 0.2 cm in each dimension. Use differentials to estimate the maximum error in calculating the surface area of the box. (Round your answer to one decimal place.)

Answer 1

The maximum error in calculating the surface area of the box is
`158.4 \:cm^2`.

Explanation:

Differentials are infinitely small quantities. Given a function
`y=f(x)` we call
`dy` and
`dx` differentials and the relationship between them is given by,

`dy=f'(x) \:dx`

If the dimensions of the box are
`l`,
`w` and
`h`, its surface area is
`A=2(wl+hl+hw)` and

`dA=(\partial A)/(\partial l)dl+(\partial A)/(\partial w)dw+(\partial A)/(\partial h)dh\\\\dA=2(w+h)dl+2(h+l)dw+2(w+l)dh`

We are given that
`|\Delta l|\leq 0.2`,
`|\Delta w|\leq 0.2`, and
`|\Delta h|\leq 0.2`.

To find the largest error in the surface area, we therefore use
`dl=0.2, dw=0.2, dh=0.2` together with
`l=97, w=67,h=34`

`dA=2(w+h)dl+2(h+l)dw+2(w+l)dh\\\\dA=2(67+34)\cdot 0.2+2(34+97)\cdot 0.2+2(67+97)\cdot 0.2\\\\dA=158.4 \:cm^2`

An error of 0.2 cm in each dimension could lead to an error of
`158.4 \:cm^2` in the calculated surface area.