Question

Two watch towers at a historic fort are located 375 m apart. The first tower is 14 m tall, and the second tower is 30 m tall.

a) What is the angle of depression from the top of the second tower to the top of the first tower?
b) The guards in the towers simultaneously spot a suspicious car parked between the towers. The angle of depression from the lower tower to the car is 7.7 degrees. The angle of depression from the higher tower is 6.3 degrees. Which guard is closer to the car? Explain how you know.

Answer 1

The angle of depression from the second tower to the first tower can be found using the tangent function. The guard with the smaller angle of depression is closer to the car.

Step-by-step explanation:

Let's start by finding the angle of depression from the top of the second tower to the top of the first tower.

To find this angle, we can use the tangent function, which is defined as the opposite side divided by the adjacent side.

Let's call the angle of depression θ and the distance between the towers D. The opposite side is the height of the second tower (30 m) and the adjacent side is the distance between the towers (375 m).

So, we have tan(θ) = 30/375. Taking the inverse tangent of both sides, we find θ = tan-1(30/375).

Now, let's move on to part b. The guard in the lower tower has an angle of depression of 7.7 degrees to the car, and the guard in the higher tower has an angle of depression of 6.3 degrees to the car.

The guard with the smaller angle is closer to the car. In this case, the guard in the higher tower is closer to the car because the angle of depression is smaller.