Differentials — Question 5

Question 5 - Solution

We are given that the diagonal of a square is $d = 10$ cm with an error of $dd = 0.2$ cm. The area of a square in terms of its diagonal is:

$$s = \frac{d}{\sqrt{2}}, \quad A = s^2 = \left(\frac{d}{\sqrt{2}}\right)^2 = \frac{d^2}{2}$$

Differentiate: $$dA = \frac{d}{dx}\left(\frac{d^2}{2}\right) \cdot dd = d \cdot dd$$

(a) Maximum error in the area: $$dA = d \cdot dd = 10 \cdot 0.2 = \boxed{2 \text{ cm}^2}$$

(b) Relative and percentage error:

Actual area: $$A = \frac{10^2}{2} = 50 \text{ cm}^2$$

Relative error: $$\frac{dA}{A} = \frac{2}{50} = 0.04$$

Percentage error: $$0.04 \times 100 = \boxed{4\%}$$

Final Answer:

• Maximum error in area: $\boxed{2 \text{ cm}^2}$

• Relative error: $\boxed{0.04}$

• Percentage error: $\boxed{4\%}$