Question

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.

Answer 1

Answer:

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