Final answer:
The angle of depression of the tunnel's descent is approximately 51.31 degrees, found by using the cosine function with the depth and slope length of the tunnel.
Step-by-step explanation:
To find the angle of depression of the tunnel's descent, we can use trigonometric functions. We know the depth of the tunnel is 52 meters (the vertical leg of a right-angled triangle) and the length of the tunnel sloping downward is 83 meters (the hypotenuse of the triangle).
We can use the cosine of the angle, which is the adjacent side (depth) divided by the hypotenuse (length of the slope). So the cosine of the angle θ is given by:
cos(θ) = depth / slope length
cos(θ) = 52 / 83
θ = cos⁻¹(52 / 83)
Now we need to calculate the angle using a calculator, and it will give us the angle of depression.
Let's calculate it:
θ = cos⁻¹(52 / 83) ≈ cos⁻¹(0.6265) ≈ 51.31° (rounded to two decimal places)
Thus, the angle of depression of the tunnel's descent is approximately 51.31 degrees.