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Answer 1

3a) Explanation of "Angle of Depression": The "angle of depression" is the angle formed below the horizontal line of sight when viewing an object. In a gorge, it's the angle from one edge to the bottom corner.

3b) Depth of the Gorge Calculation: The gorge depth is approximately 199.51 m.

4a) Distance from Lighthouse Now: After sailing north, the ship is now approximately 10.47 km away from the lighthouse.

4b) Distance LT Calculation: Using the Pythagorean Theorem, the distance LT is approximately 22.10 km.

3a) Explanation of "Angle of Depression":

The "angle of depression" refers to the angle formed between the line of sight from an observer's viewpoint to an object or point below the observer's horizontal line of sight. In the context of a gorge with a rectangular cross-section, it is the angle formed between the horizontal line of sight from one edge of the gorge to the bottom corner below.

3b) Depth of the Gorge Calculation:

Given that the gorge is 60 m wide and the angle of depression is 72 degrees, we can use trigonometry to find the depth (d) of the gorge. The tangent of the angle of depression is the ratio of the opposite side (depth) to the adjacent side (width):

`\[ \tan(72^\circ) = (d)/(60) \]`

Solving for d:

`\[ d = 60 \cdot \tan(72^\circ) \]`

`\[ d \approx 199.51 \, \text{m} \]`

4a) Distance from Lighthouse Now:

After sailing 19.5 km north to point S, the ship is forming a right-angled triangle TSL. To find the distance from the ship to the lighthouse (LS), we can use the sine of the angle TSL:

`\[ \sin(33^\circ) = (x)/(19.5) \]`

Solving for x:

`\[ x = 19.5 \cdot \sin(33^\circ) \]`

`\[ x \approx 10.47 \, \text{km} \]`

4b) Distance LT Calculation using Pythagorean Theorem:

To find the distance LT, we use the Pythagorean Theorem in triangle TSL:

`\[ LT^2 = ST^2 + SL^2 \]`

`\[ LT = √(19.5^2 + 10.47^2) \]`

`\[ LT \approx 22.10 \, \text{km} \]`