An engineer measures the distance along the ground to
the base of a tree to be 12m to the nearest
metre. If the
angle of elevation to the top of the tree is 40" to the
nearest degree, find
a the upper bound for the tree height
b the lower bound for the tree height



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