Answer :
To find the upper and lower bounds for the height of the tree, we'll use the given measurements and the concept of significant figures and angles.
Given:
- Distance to the base of the tree (ground to tree): \( d = 12 \) m (measured to the nearest metre).
- Angle of elevation to the top of the tree: \( \theta = 40^\circ \) (measured to the nearest degree).
### Step 1: Upper Bound for Tree Height
The upper bound for the tree height \( h \) occurs when we consider the maximum possible height given the measured distance \( d \) and angle \( \theta \).
1. **Convert angle to radians for accuracy:**
\[ \theta_{\text{rad}} = \frac{40 \pi}{180} = \frac{2}{9} \pi \]
2. **Calculate the upper bound for height \( h \):**
\[ h_{\text{upper}} = d \cdot \tan(\theta_{\text{rad}} + \frac{\pi}{2}) \]
Here, \( \tan(\theta_{\text{rad}} + \frac{\pi}{2}) \) gives us the maximum possible height.
Plugging in the values:
\[ h_{\text{upper}} = 12 \cdot \tan\left(\frac{2}{9} \pi + \frac{\pi}{2}\right) \]
Calculate \( \tan\left(\frac{2}{9} \pi + \frac{\pi}{2}\right) \):
\[ \tan\left(\frac{2}{9} \pi + \frac{\pi}{2}\right) = \tan\left(\frac{5}{9} \pi\right) \]
Using the tangent function:
\[ \tan\left(\frac{5}{9} \pi\right) \approx \tan\left(\frac{\pi}{2} - \frac{4}{9} \pi\right) = \cot\left(\frac{4}{9} \pi\right) \]
Calculate the cotangent value and then multiply by 12 meters
