The dimensions of a closed rectangular box are measured as 50 centimeters, 60 centimeters, and 70 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.

the maximum error in calculating the surface area of the box is 152 cm2

Explanation:

Assuming he dimensions of the box is l, w and h (for length, width and height). The surface area is then: S (l, w, h) = 2lw + 2wh + 2lh = 2 (lw + wh + lh)

The change in area can be written as: ∆S ≈ dS = Sl dl + Sw dw + Sh dh

where the partial derivatives are evaluated at l = 80, w = 60 and h = 50, and

dl = dw = dh = 0.2.

The partial derivatives are computed:

Sl = 2 (w + h) = 220 Sw = 2 (l + h) = 260 Sh = 2 (l + w) = 280

Substituting these in for dS,

dS = 220 · 0.2 + 260 · 0.2 + 280 · 0.2 = 152 cm2