Final answer:
Using trigonometry and the change in the angle of depression, we can calculate the car's speed and the time it will take to reach the base of the observation tower from the position when the angle of depression was 45 degrees.
Step-by-step explanation:
Calculating Time for a Car to Reach an Observation Tower
Assuming the car maintains a uniform speed and the observation tower's height remains constant, the angles of depression and the car's distance from the base of the tower can form right-angled triangles. When the angle of depression changes from 30 degrees to 45 degrees, the car has moved from a position where it forms two distinct triangles with the tower. At 45 degrees, we know that the distance from the base of the tower to the car is equal to the height of the tower since tan(45 degrees) is 1. Let's denote the height of the tower as h and the initial position's distance when the angle is 30 degrees as d. Using trigonometry, tan(30 degrees)=h/d, which can be solved to find the initial distance d.
Given that it took the car 12 minutes to change the angle of depression from 30 degrees to 45 degrees, we can calculate the distance the car covered and hence the distance per minute. Once we know the distance per minute, we can calculate how soon the car will cover the distance equal to the height of the tower (h) and reach the base of the tower, which occurs when the angle of elevation from the car to the top of the tower is 90 degrees.