Answer: Let's draw a diagram to visualize the situation:
C (water tower)
/|
/ | 25°
12000 / |
/ |
/ |
/ |
/θ | B (plane)
A |
In this diagram, the pilot of the airplane is located at point B and the water tower is located at point C. The angle of depression from the airplane to the base of the tower is 25°. We are asked to find the length of the line of sight from the airplane to the tower, which is the distance AC.
We can use trigonometry to solve for AC. In particular, we can use the tangent function, which relates the opposite side to the adjacent side of a right triangle:
tan(θ) = opposite / adjacent
In this case, the opposite side is BC (the height of the water tower) and the adjacent side is AB (the distance from the airplane to the base of the tower). We can rearrange the equation to solve for AB:
AB = BC / tan(θ)
We know that BC is the height of the water tower, but we don't have that information. However, we can use the fact that the angle of depression is 25° to find BC. The angle of depression is the angle between the horizontal line (which we can assume is the same as the ground level) and the line of sight from the airplane to the base of the tower. Therefore, the angle between the line of sight and the vertical line (which is perpendicular to the ground) is 90° - 25° = 65°. This means that the triangle ABC is a right triangle, with angle θ = 65°.
Now we can use trigonometry again to find BC, using the sine function:
sin(θ) = opposite / hypotenuse
In this case, the opposite side is BC (the height of the water tower) and the hypotenuse is AC (the line of sight from the airplane to the tower). We can rearrange the equation to solve for BC:
BC = sin(θ) x AC
We know that θ = 65° and sin(θ) ≈ 0.9063 (you can use a calculator to find this value). Substituting these values into the equation gives us:
BC = 0.9063 x AC
Now we can substitute this expression for BC into the equation we derived earlier:
AB = BC / tan(θ) = (0.9063 x AC) / tan(65°)
We can simplify this expression by noting that tan(65°) ≈ 2.1445 (you can use a calculator to find this value). Substituting this value gives us:
AB = (0.9063 x AC) / 2.1445
Multiplying both sides by 2.1445 gives us:
2.1445 x AB = 0.9063 x AC
Dividing both sides by 0.9063 gives us:
AC = (2.1445 x AB) / 0.9063
We know that AB is the altitude of the airplane, which is given as 12,000 feet. Substituting this value gives us:
AC = (2.1445 x 12,000) / 0.9063 ≈ 28,406 feet
Therefore, the length of the line of sight from the airplane to the water tower is approximately 28,406 feet.
Explanation: