1. **Problem statement:** We have a right triangle with one side length $9.5$ inches and an opposite angle of $26^\circ 45'$ with a possible error of $15'$. (a) We want to approximate the percent error (in radians) in computing the length of the hypotenuse. (b) We want to estimate the maximum allowable percent error (in radians) in measuring the angle if the error in computing the hypotenuse length cannot exceed 3%. 2. **Formulas and rules:** - The hypotenuse $h$ can be found using the sine function: $$h = \frac{\text{opposite side}}{\sin(\theta)} = \frac{9.5}{\sin(\theta)}$$ - Percent error in $h$ due to error in $\theta$ can be approximated using differentials: $$\frac{\Delta h}{h} \approx \left| \frac{d}{d\theta} \left( \frac{9.5}{\sin(\theta)} \right) \right| \Delta \theta \div h$$ - Since $h = \frac{9.5}{\sin(\theta)}$, the derivative is: $$\frac{d h}{d \theta} = 9.5 \cdot \frac{-\cos(\theta)}{\sin^2(\theta)} = -h \cot(\theta)$$ - Therefore, relative error in $h$ is: $$\frac{\Delta h}{h} \approx | -\cot(\theta) | \Delta \theta = \cot(\theta) \Delta \theta$$ 3. **Convert angle and error to radians:** - $26^\circ 45' = 26 + \frac{45}{60} = 26.75^\circ$ - Convert to radians: $$\theta = 26.75 \times \frac{\pi}{180} \approx 0.4669$$ - Error in angle $15' = \frac{15}{60} = 0.25^\circ$ - Convert error to radians: $$\Delta \theta = 0.25 \times \frac{\pi}{180} \approx 0.00436$$ 4. **Calculate cotangent:** $$\cot(\theta) = \frac{\cos(0.4669)}{\sin(0.4669)} \approx \frac{0.8936}{0.4504} \approx 1.984$$ 5. **(a) Percent error in hypotenuse:** $$\frac{\Delta h}{h} \approx 1.984 \times 0.00436 = 0.00865 = 0.865\%$$ Rounded to two decimals: **0.87%** 6. **(b) Maximum allowable percent error in angle if hypotenuse error $\leq 3\%$:** We have: $$0.03 = \cot(\theta) \Delta \theta_{max}$$ Solve for $\Delta \theta_{max}$: $$\Delta \theta_{max} = \frac{0.03}{\cot(\theta)} = \frac{0.03}{1.984} \approx 0.01512$$ 7. **Final answers:** - (a) Percent error in hypotenuse $\approx 0.87\%$ - (b) Maximum allowable percent error in angle $\approx 0.01512$ radians These results show how small errors in angle measurement affect the hypotenuse length in a right triangle.