From a small boat the sea the angle of elevation to the top of a cliff, 128m above sea level, is 28 degrees. After rowing directly towards the cliff, it's angle of elevation from the boat is found to be 46 degrees. How far did the boat travel while being towed towards the cliff? ( Show with diagram )

Answer 1

Sure, here's the diagram:

```
|\
| \
| \ 28°
128m| \
| \
| \
| \
|46° \
| \
Boat's | \
original | \
position |__________\
x
```

Let's call the distance the boat travelled towards the cliff "x". We can use trigonometry to solve for it.

From the first angle of elevation, we know that:

```
tan(28°) = 128 / distance from boat to cliff
```

We can rearrange this to solve for the distance:

```
distance from boat to cliff = 128 / tan(28°)
```

Now, after rowing towards the cliff, the boat sees a new angle of elevation of 46 degrees. From this, we know:

```
tan(46°) = 128 / (distance from boat to cliff - x)
```

We can solve for x by substituting the first equation into the second equation:

```
tan(46°) = 128 / (128 / tan(28°) - x)
```

Simplifying this, we get:

```
x = (128 / tan(28°)) - (128 / tan(46°))
```

Plugging in the values, we get:

```
x = (128 / 0.531) - (128 / 1.035)
x = 196.25 - 123.89
x = 72.36
```

So the boat travelled 72.36 meters towards the cliff.

Answer 2

As always, draw a diagram, and review your basic trig functions.
It got everything correct, right up to

x = (128 / tan(28°)) - (128 / tan(46°))
x = (128 / 0.531) - (128 / 1.035)
and then just plain blew it!
But the result is 117.12 m