1. **Problem statement:** Calculate the percentage uncertainty in the experimental result $$y = \frac{x_1 x_2}{x_3}$$ given uncertainties $$W_{x_1} = \pm 1.0$$, $$W_{x_2} = \pm 0.5$$, $$W_{x_3} = 0.2$$ and measurement ranges $$5 < x_1 < 100$$, $$5 < x_2 < 100$$, $$15 < x_3 < 100$$. 2. **Formula for uncertainty propagation:** For a function $$y = \frac{x_1 x_2}{x_3}$$, the relative (percentage) uncertainty in $$y$$ is given by: $$\frac{W_y}{y} = \sqrt{\left(\frac{W_{x_1}}{x_1}\right)^2 + \left(\frac{W_{x_2}}{x_2}\right)^2 + \left(\frac{W_{x_3}}{x_3}\right)^2}$$ 3. **Calculate percentage uncertainty at lower limits:** - Use $$x_1 = 5$$, $$x_2 = 5$$, $$x_3 = 15$$ - Calculate relative uncertainties: $$\frac{W_{x_1}}{x_1} = \frac{1.0}{5} = 0.2 = 20\%$$ $$\frac{W_{x_2}}{x_2} = \frac{0.5}{5} = 0.1 = 10\%$$ $$\frac{W_{x_3}}{x_3} = \frac{0.2}{15} \approx 0.0133 = 1.33\%$$ - Combine: $$\frac{W_y}{y} = \sqrt{0.2^2 + 0.1^2 + 0.0133^2} = \sqrt{0.04 + 0.01 + 0.000177} = \sqrt{0.050177} \approx 0.224\ (22.4\%)$$ 4. **Calculate percentage uncertainty at upper limits:** - Use $$x_1 = 100$$, $$x_2 = 100$$, $$x_3 = 100$$ - Calculate relative uncertainties: $$\frac{W_{x_1}}{x_1} = \frac{1.0}{100} = 0.01 = 1\%$$ $$\frac{W_{x_2}}{x_2} = \frac{0.5}{100} = 0.005 = 0.5\%$$ $$\frac{W_{x_3}}{x_3} = \frac{0.2}{100} = 0.002 = 0.2\%$$ - Combine: $$\frac{W_y}{y} = \sqrt{0.01^2 + 0.005^2 + 0.002^2} = \sqrt{0.0001 + 0.000025 + 0.000004} = \sqrt{0.000129} \approx 0.01136\ (1.14\%)$$ 5. **Final answers:** - Percentage uncertainty in $$y$$ at lower limits is approximately **22.4%**. - Percentage uncertainty in $$y$$ at upper limits is approximately **1.14%**. This shows that the relative uncertainty decreases as the measured values increase because the absolute uncertainties remain constant while the measured values grow.