Final answer:
To find the angle that the normal to the surface of the mirror should make with due south, we can use trigonometry. Using the Pythagorean theorem, we can find the length of BC. Then, using the sine function, we can find the angle of 23.5° with due south.
Step-by-step explanation:
To find the angle that the normal to the surface of the mirror should make with due south, we need to use trigonometry. We can use the properties of right triangles to solve this problem.
First, we can draw a diagram to represent the situation. The laser is located at point A, the mirror is located at point B, and the detector is located at point C. We are given that AB = 47.0 km and AC = 119 m.
Next, we can use the Pythagorean theorem to find the length of BC:
BC^2 = AC^2 + AB^2.
Plugging in the values, we get BC^2 = (119 m)^2 + (47.0 km)^2.
Simplifying, we find BC^2 = 14,081,000,000 m^2.
Taking the square root, we find BC ≈ 118,756 m.
Now, we can use the sine function to find the angle. The sine of an angle is equal to the ratio of the length of the opposite side to the length of the hypotenuse. In this case, the opposite side is AB and the hypotenuse is BC.
So, sin(θ) = AB / BC. Plugging in the values, we find sin(θ) = 47.0 km / 118,756 m.
Simplifying, we find sin(θ) ≈ 0.3966.
Taking the inverse sine, we find θ ≈ 23.5°.
Therefore, the normal to the surface of the mirror should make an angle of approximately 23.5° with due south.