Question

The dimensions of a closed rectangular box are measured as 60 centimeters, 50 centimeters, and 70 centimeters, with an error in each measurement of at most 0.2 centimeters. Use differentials to estimate the maximum error in calculating the surface area of the box.

Answer 1

The maximum error in calculating the surface area of the box is 72 square centimeters.

Explanation:

From Geometry, the surface area of the closed rectangular box (
`A_(s)`), in square centimeters, is represented by the following formula:

`A_(s) = w\cdot l + (w + l)\cdot h` (1)

Where:

`w` - Width, in centimeters.

`l` - Length, in centimeters.

`h` - Height, in centimeters.

And the maximum error in calculating the surface area (
`\Delta A_(s)`), in square centimeters, is determined by the concept of total differentials, used in Multivariate Calculus:

`\Delta A_(s) = \left(l+h\right)\cdot \Delta w + \left(w+h\right)\cdot \Delta l + (w+l)\cdot \Delta h` (2)

Where:

`\Delta w` - Measurement error in width, in centimeters.

`\Delta l` - Measurement error in length, in centimeters.

`\Delta h` - Measurement error in height, in centimeters.

If we know that
`\Delta w = \Delta h = \Delta l = 0.2\,cm`,
`w = 60\,cm`,
`l = 50\,cm` and
`h = 70\,cm`, then the maximum error in calculating the surface area is:

`\Delta A_(s) = (120\,cm + 130\,cm + 110\,cm)\cdot (0.2\,cm)`

`\Delta A_(s) = 72\,cm^(2)`

The maximum error in calculating the surface area of the box is 72 square centimeters.