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 )
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
To solve this problem, we can use trigonometry. Let's start by drawing a diagram:
```
|
| *
| /|
Sea | / |
| c / |
| / |
| / |
| / |
|/ | Cliff
---------------------------
a b
```
In the diagram:
- "Sea" represents the sea level.
- "*" represents the boat's initial position.
- "Cliff" represents the top of the cliff.
- "a" represents the distance the boat traveled.
- "b" represents the perpendicular distance from the boat to the base of the cliff.
- "c" represents the height of the cliff, which is given as 128m.
We are given that the angle of elevation from the boat to the top of the cliff is 28 degrees. This means the angle between the line connecting the boat to the cliff base and the horizontal axis is also 28 degrees.
Using trigonometry, we can write:
tan(28°) = b / a
Similarly, when the boat moves toward the cliff, we are given that the angle of elevation is 46 degrees. This means the angle between the line connecting the boat to the cliff base and the horizontal axis is also 46 degrees.
Using trigonometry, we can write:
tan(46°) = (b + c) / a
Since we want to find the distance the boat traveled, we can solve these two equations simultaneously for "a".
First, let's solve the first equation for "b":
b = a * tan(28°)
Now, substitute this value into the second equation:
tan(46°) = (a * tan(28°) + c) / a
Simplifying:
tan(46°) = tan(28°) + (c / a)
Now, let's solve this equation for "a":
a = c / (tan(46°) - tan(28°))
Substitute the given values:
a = 128 / (tan(46°) - tan(28°))
Calculating the values:
a ≈ 128 / (1.0353 - 0.5317)
a ≈ 128 / 0.5036
a ≈ 254.245
Therefore, the boat traveled approximately 254.245 meters while being towed towards the cliff.
To solve this problem, we can use trigonometry, specifically the tangent function.
Let's set up the diagram:
```
* Cliff
/|
/ |
/ | 128m
/ |
/ θ1 = 28°
*----*
Boat
```
The angle of elevation to the top of the cliff from the boat, θ1, is 28 degrees, and the height of the cliff, AC, is 128m.
Next, let's update the diagram after the boat has moved towards the cliff:
```
* Cliff
/|
/ |
/ | 128m
/ |
/ * P
/ /|
/ / |
/ θ2 |
*----* B
```
The angle of elevation to the top of the cliff from the new position of the boat, θ2, is 46 degrees.
Now, we need to find the distance the boat (BP) traveled while being towed towards the cliff.
Let's consider the right triangle BCP.
Using the tangent function, we can write:
tan(θ1) = AC / BP (equation 1)
And
tan(θ2) = AC / (BP + x) (equation 2)
where x is the distance from B to P.
Now, let's solve for BP by rearranging equation 1:
BP = AC / tan(θ1) (equation 3)
Using equation 3, we can substitute BP in equation 2:
tan(θ2) = AC / (AC / tan(θ1) + x)
Simplifying further:
tan(θ2) = tan(θ1) / (1 - tan(θ1) * x / AC)
Next, isolate x:
tan(θ2) * AC = tan(θ1) * x - tan(θ1) * x^2
Rearranging:
tan(θ1) * x^2 - tan(θ1) * x + tan(θ2) * AC = 0
This is a quadratic equation in x. We can solve for x using the quadratic formula:
x = [ -b ± √(b^2 - 4ac) ] / (2a)
For our equation:
a = tan(θ1)
b = -tan(θ1)
c = tan(θ2) * AC
Now, substitute the values and solve for x:
θ1 = 28°
θ2 = 46°
AC = 128m
a = tan(28°)
b = -tan(28°)
c = tan(46°) * 128
Plug these values into the quadratic formula to calculate x.
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.