To estimate the error in the hypotenuse length using a linear approximation, we need to apply differential calculus to the function for the hypotenuse of a right triangle.

Step 1: Hypotenuse Formula

The hypotenuse $c$ of a right triangle with side lengths $a$ and $b$ is given by the Pythagorean theorem: $c = \sqrt{a^2 + b^2}$ Here, $a = 5 \, \text{cm}$ and $b = 12 \, \text{cm}$.

Step 2: Calculate the Hypotenuse

Using the given side lengths $a = 5 \, \text{cm}$ and $b = 12 \, \text{cm}$: $c = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \, \text{cm}$

Step 3: Error in the Hypotenuse

The error in the hypotenuse $\Delta c$ is approximated by differentiating the hypotenuse formula with respect to $a$ and $b$: $dc = \frac{\partial c}{\partial a} da + \frac{\partial c}{\partial b} db$ The partial derivatives are: $\frac{\partial c}{\partial a} = \frac{a}{\sqrt{a^2 + b^2}}, \quad \frac{\partial c}{\partial b} = \frac{b}{\sqrt{a^2 + b^2}}$ Now, plugging in the values for $a = 5$, $b = 12$, and $c = 13$: $\frac{\partial c}{\partial a} = \frac{5}{13}, \quad \frac{\partial c}{\partial b} = \frac{12}{13}$

The errors in $a$ and $b$ are given as $\Delta a = 0.1 \, \text{cm}$ and $\Delta b = 0.2 \, \text{cm}$. So the error in the hypotenuse is: $\Delta c \approx \frac{5}{13} \times 0.1 + \frac{12}{13} \times 0.2$ Now, let's calculate this.

Step 4: Calculation of Hypotenuse ErrorThe estimated maximum error in the hypotenuse length is approximately 0.22 cm (rounded to two decimal places).

Would you like more details or have any questions?

Here are some related questions you could explore:

1. How do you estimate the error in the area of the triangle?
2. What are the applications of error propagation in physics and engineering?
3. How would the maximum error change if the measurement errors were smaller?
4. How does this approximation compare with more advanced error analysis techniques?
5. What is the role of partial derivatives in error propagation?

Tip: When calculating errors, always double-check that the units and rounding conventions are consistent to avoid misinterpretation.