An owl is 7 metres above ground level, in a tree - HSC - SSCE Mathematics Standard - Question 12 - 2019 - Paper 1

To find the straight line distance the owl must fly to catch the mouse, we can use the relationship defined by the tangent of the angle of depression:

1. Let the height of the owl above the ground be represented as 'h' and the angle of depression as 'θ'. Here, h = 7 meters and θ = 32°.

2. We set up the relationship using the tangent function:

   $\tan(\theta) = \frac{\text{height}}{\text{horizontal distance}}$

   Rearranging gives:

   $\text{horizontal distance} = \frac{\text{height}}{\tan(\theta)}$

   Thus, horizontal distance = ( \frac{7}{\tan(32°)} ).

3. Calculate the horizontal distance:

   $\text{horizontal distance} \approx \frac{7}{0.6249} \approx 11.2 \text{ m}$

4. Now, to find the straight-line distance, use the Pythagorean theorem:

   $d = \sqrt{h^2 + (\text{horizontal distance})^2} = \sqrt{7^2 + 11.2^2}$

   $\approx \sqrt{49 + 125.44} \approx \sqrt{174.44} \approx 13.2 \text{ m}$.

Thus, the straight-line distance that the owl must fly is approximately 13.2 meters.