At Westonci.ca, we provide clear, reliable answers to all your questions. Join our vibrant community and get the solutions you need. Our Q&A platform provides quick and trustworthy answers to your questions from experienced professionals in different areas of expertise. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform.
Sagot :
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
Thank you for choosing our service. We're dedicated to providing the best answers for all your questions. Visit us again. We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. Thank you for visiting Westonci.ca. Stay informed by coming back for more detailed answers.
