A car is being driven, in a straight line and at a uniform speed, towards the base of a vertical tower of height 30 feet. The top of the tower is observed from the car and, in the process, the angle of elevation changes from `45^(@)` at B to `60^(@)` at A. What is the distance, in feet, to the nearest integer, between the points A and B?

Answer

To solve the problem, we will use trigonometric functions to find the distances from points A and B to the base of the tower. Here’s a step-by-step solution:

### Step 1: Understand the Problem
We have a vertical tower of height 30 feet. The angle of elevation from point B is 45 degrees and from point A is 60 degrees. We need to find the distance between points A and B.

### Step 2: Draw a Diagram
Draw a vertical line representing the tower (CD) with a height of 30 feet. Mark point A and point B on the ground, where point A is closer to the tower than point B. The angles of elevation from these points to the top of the tower (C) are 60 degrees at A and 45 degrees at B.

### Step 3: Set Up the Trigonometric Relationships
Using the tangent function, we can relate the height of the tower to the distances from points A and B to the base of the tower (D).

1. For point B (angle of elevation = 45 degrees):
\[ \tan(45^\circ) = \frac{CD}{BD} \]
Since \(\tan(45^\circ) = 1\):
\[ 1 = \frac{30}{BD} \Rightarrow BD = 30 \text{ feet} \]

2. For point A (angle of elevation = 60 degrees):
\[ \tan(60^\circ) = \frac{CD}{AD} \]
Since \(\tan(60^\circ) = \sqrt{3}\):
\[ \sqrt{3} = \frac{30}{AD} \Rightarrow AD = \frac{30}{\sqrt{3}} = 10\sqrt{3} \text{ feet} \]

### Step 4: Calculate the Distance Between Points A and B
The distance between points A and B is the difference between the distances from each point to the base of the tower:
\[ AB = BD - AD \]
Substituting the values we found:
\[ AB = 30 - 10\sqrt{3} \]

### Step 5: Calculate the Numerical Value
Now we will calculate \(10\sqrt{3}\):
\[ \sqrt{3} \approx 1.732 \Rightarrow 10\sqrt{3} \approx 17.32 \]
Thus,
\[ AB \approx 30 - 17.32 = 12.68 \]

### Step 6: Round to the Nearest Integer
Finally, we round 12.68 to the nearest integer:
\[ \text{Distance} \approx 13 \text{ feet} \]

### Final Answer
The distance between points A and B is approximately **13 feet**.