This problem is asking us to determine how accurately the angle of elevation must be measured to ensure that the percentage error in estimating the height of the building remains below 3%. Let’s break it down step by step.

Given:

• Distance from the base of the building = $10 \, \text{ft}$
• Measured angle of elevation = $5^\circ$
• Acceptable percentage error = $3\%$

Key relationships:

1. Height of the building ($h$): Using trigonometry: $h = 10 \cdot \tan(\theta)$ where $\theta$ is the angle of elevation.

2. Error propagation: The error in the height due to an error in the angle is calculated using the derivative of $h$ with respect to $\theta$: $\frac{dh}{h} = \sec^2(\theta) \cdot d\theta$ Percentage error in $h$: $\text{Percentage error} = \left( \sec^2(\theta) \cdot d\theta \right) \cdot 100$

3. Condition: To keep the error below $3\%$, we set: $\sec^2(\theta) \cdot d\theta \cdot 100 < 3$ Simplify: $d\theta < \frac{3}{100 \cdot \sec^2(\theta)}$

Step-by-step Calculation:

1. Compute $\sec^2(5^\circ)$: $\sec^2(5^\circ) = 1 + \tan^2(5^\circ)$ Using $\tan(5^\circ) \approx 0.08749$: $\sec^2(5^\circ) = 1 + (0.08749)^2 \approx 1.00765$

2. Plug $\sec^2(5^\circ)$ into the inequality: $d\theta < \frac{3}{100 \cdot 1.00765} \approx \frac{3}{100.765} \approx 0.02976 \, \text{radians}$

3. Convert $d\theta$ to degrees: $d\theta \, (\text{degrees}) = 0.02976 \cdot \frac{180}{\pi} \approx 1.705^\circ$

Final Answer:

The angle of elevation must be measured with an accuracy of approximately $\pm 1.7^\circ$ to ensure the percentage error in estimating the height is less than 3%.

Would you like a more detailed explanation of any step or further assistance? Here are five related questions to explore:

1. How would the required accuracy change if the angle of elevation were larger, say $10^\circ$?
2. What happens to the error if the distance from the building increases?
3. How is the calculation affected if the percentage error tolerance is tightened to 1%?
4. Can this formula be generalized to any angle and distance?
5. How would rounding errors affect the measurement and error calculation?

Tip: Always ensure that trigonometric calculations are done with appropriate units (radians vs degrees).