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The length and width of a rectangle are measured as 31 cm and 28 cm, respectively, with an error in measurement of at most 0.1 cm in each. Use differentials to estimate the maximum error in the calculated area of the rectangle.

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Answer:

[tex]\Delta A = 5.9\,cm^{2}[/tex]

Step-by-step explanation:

The area of an rectangle is given by the following formula:

[tex]A = w\cdot h[/tex]

Where:

[tex]w[/tex] - Width, in centimeters.

[tex]h[/tex] - Height, in centimeters.

The differential of the expression is derived hereafter:

[tex]\Delta A = \frac{\partial A}{\partial w} \cdot \Delta w + \frac{\partial A}{\partial h}\cdot \Delta h[/tex]

[tex]\Delta A = h \cdot \Delta w + w \cdot \Delta h[/tex]

[tex]\Delta A = (31\,cm)\cdot (0.1\,cm) + (28\,cm)\cdot (0.1\,cm)[/tex]

[tex]\Delta A = 5.9\,cm^{2}[/tex]

fichoh

Using differentials the maximum error in the calculated area of the rectangle wi’ould be 5.9 cm

The area formular of a rectangle is :

  • Area = Length(l) × width(w)
  • w = 28 cm
  • l = 31 cm
  • Error, Δe = 0.1cm

Maximum error can be defined thus :

  • Δmax = (L × Δe) + (W × Δe)

Δmax = (L × Δe) + (W × Δe)

Δmax = (31 × 0.1) + (28 × 0.1)

Δmax = 3.1 + 2.8

Δmax = 5.9 cm

Hence, the maximum error in the calculated area value is 5.9 cm.

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