Question

taranaki maunga is approximately conical and is part of egmont national park.

• it is 2518 metres high
• the radius to the edge of egmont national park is 9.6 km.

inglewood high school is 9.17 km from the edge of egmont national park.

below lists information on the three most popular walks in egmont national park:
\

$$\begin{tabular}{|l|l|l|} \\hline name & distance of track & time to complete \\\\ \\hline mount taranaki track & 11.7 km & 6 hours 6 minutes \\\\ \\hline pouakai tarns & 12.2 km & 4 hours 48 minutes \\\\ \\hline pouakai circuit & 24.5 km & 9 hours 17 minutes \\\\ \\hline \\end{tabular}$$

Explanation:

🆕 New Concept Discovered: Right-Angled Triangles in 3D Space
Using 3D geometry to find direct line-of-sight distances.

Step 1: Identify the dimensions and units

To find the direct line-of-sight distance from the peak of Taranaki Maunga to Inglewood High School, we can model the scenario as a right-angled triangle in 3D space.

First, let's convert all measurements to the same unit (kilometers):

• Height of the mountain peak, \( h = 2518\text{ m} = 2.518\text{ km} \)
• Radius of the park base, \( r = 9.6\text{ km} \)
• Distance from the park boundary to Inglewood High School, \( d = 9.17\text{ km} \)

Step 2: Calculate the total horizontal distance

The horizontal distance \( D \) from the center of the mountain's base (directly beneath the peak) to Inglewood High School is the sum of the park's radius and the distance from the park's edge to the school:

$$ D = r + d $$

$$ D = 9.6\text{ km} + 9.17\text{ km} = 18.77\text{ km} $$

Step 3: Apply the Pythagorean theorem

The peak of the mountain, the center of the base, and Inglewood High School form a right-angled triangle where:

• The vertical leg is the height of the peak: \( h = 2.518\text{ km} \)
• The horizontal leg is the total ground distance: \( D = 18.77\text{ km} \)
• The hypotenuse \( L \) is the direct line-of-sight distance.

Using the Pythagorean theorem:

$$ L = \sqrt{D^2 + h^2} $$

$$ L = \sqrt{18.77^2 + 2.518^2} $$

$$ L = \sqrt{352.3129 + 6.340324} $$

$$ L = \sqrt{358.653224} $$

$$ L \approx 18.94\text{ km} $$

Answer:

The direct line-of-sight distance from the peak of Taranaki Maunga to Inglewood High School is approximately \( 18.94\text{ km} \) (or \( 18,940\text{ m} \)).