The dimensions of a closed rectangular box are measured as 8

The dimensions of a closed rectangular box are measured as 88 cm, 52 cm, and 40 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.)

___ cm^2

Solution

The surface area of a box of length x, width y and height z is S = 2(xy + yz + xz).

The differential of S is dS(x, y, z) = S_x(x, y, z)dx+S_y(x, y, z)dy+S_z(x, y, z)dz = 2(y+z)dx+2(x+z)dy+2(x+y)dz.

Thus, at the point (80, 60, 50),

we have dS(88, 52, 40) = 194dx + 236dy + 280dz.

Since the maximum error in the measurement of each dimension is 0.2,

we use dx = 0.2, dy = 0.2 and dz = 0.2.

The maximum error in the measurement of the surface area is

dS = 194*0.2 + 236*0.2 + 280*0.2 = 142 sq cm.