Differentials — Question 7

Question 7 - Solution

We are given that $a = 5$ is fixed and $b = 12$ with a possible error $db = 0.1$. The hypotenuse is: $$c = \sqrt{a^2 + b^2}$$

Differentiate both sides: $$dc = \frac{1}{2}(a^2 + b^2)^{-1/2} \cdot 2b \, db = \frac{b}{\sqrt{a^2 + b^2}} \cdot db$$

Substitute known values: $$a = 5, \quad b = 12, \quad db = 0.1$$ $$c = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13$$

$$dc = \frac{12}{13} \cdot 0.1 \approx \boxed{0.0923}$$

(a) Maximum error: $$\boxed{dc \approx 0.0923 \text{ cm}}$$

(b) Relative and percentage error: $$\text{Relative error} = \frac{dc}{c} = \frac{0.0923}{13} \approx 0.0071$$ $$\text{Percentage error} = 0.0071 \times 100 \approx \boxed{0.71\%}$$

Final Answers:

• Maximum error in hypotenuse: $\boxed{0.0923 \text{ cm}}$

• Relative error: $\boxed{0.0071}$

• Percentage error: $\boxed{0.71\%}$