A man standing on top of a tower sees a car coming towards the tower. If it takes 20 minutes for the angle of depression to change from 30° to 60°, what is the time remaining for the car to reach the tower?a)20√3 minutesb)10 minutesc)10√3 minutesd)5 minutesCorrect answer is option 'B'. Can you explain this answer?

Question

A man standing on top of a tower sees a car coming towards the tower. If it takes 20 minutes for the angle of depression to change from 30&deg; to 60&deg;, what is the time remaining for the car to reach the tower?a)20&radic;3 minutesb)10 minutesc)10&radic;3 minutesd)5 minutesCorrect answer is option 'B'. Can you explain this answer?

Answer

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Answer

Answer

° to 60°, how long will it take for the car to reach the tower?

We can use trigonometry to solve this problem. Let's draw a diagram to help us visualize the situation:

```
C
|\
| \
| \ h
| \
| \
| \
| \
A-------B
d
```

In this diagram, A is the man standing on the tower, B is the base of the tower, C is the car, and h is the height of the tower. We want to find the distance d from the car to the base of the tower.

We know that the angle of depression from A to C changes from 30° to 60° as the car approaches the tower. This means that the angle ACB is also changing, from 60° to 30°. We can use the tangent function to relate the height of the tower to the distance from the car:

```
tan(60°) = h / d1 (when the angle is 60°)
tan(30°) = h / d2 (when the angle is 30°)
```

We want to find the time it takes for the car to travel from its initial distance d1 to the tower to its final distance d2 to the tower. Let's call this time t.

We know that the car is moving at a constant speed, so we can use the formula:

```
distance = speed x time
```

The distance that the car travels is the difference between its initial distance and its final distance:

```
distance = d1 - d2
```

The speed of the car is the distance it travels divided by the time it takes:

```
speed = (d1 - d2) / t
```

We can substitute the expressions for d1 and d2 in terms of h:

```
speed = (h / tan(60°) - h / tan(30°)) / t
```

Simplifying the expression using the tangent addition formula:

```
speed = h * (sqrt(3) - 1) / (3 * t)
```

Now we need to find h, the height of the tower. We can use the fact that the angle of depression is 30° when the car is at its closest distance to the tower. This means that the distance from the man to the car at that point is equal to the height of the tower:

```
h = d2 = d1 - distance = d1 - speed x t
```

Substituting the expression for speed:

```
h = d1 - h * (sqrt(3) - 1) / (3 * t) (when the angle is 30°)
```

Solving for h:

```
h + h * (sqrt(3) - 1) / (3 * t) = d1
h * (1 + sqrt(3) - 1) / (3 * t) = d1 - h
h = (3 * t * d1) / (1 + sqrt(3))
```

Now we can substitute this expression for h in the expression for speed:

```
speed = h * (sqrt(3) - 1) / (3 * t)
= (3 * t * d

Answer

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