A man on a 135 feet cliff perpendicular (at 90 degrees) to the ground looks down at an
angle of 16 degrees and sees his friend. How far away is the man from his friend? How far away
is the friend from the base of the cliff? Sketch the triangle that matches this scenario with a ruler
and protractor, and solve for the distance between the man and his friend and for the distance
between the friend and the base of the cliff.
sin 15 = 135/ hypotenuse
tan 15 = 135/d along ground
To solve this problem, we can use trigonometry. Let's start by sketching the triangle that represents the scenario:
```
|
|
|
| 135 ft
|
|------------M
| 16°
|
|-----------------------B
x ft
```
In the above sketch, "M" represents the man on the cliff, "B" represents the base of the cliff, and "x" represents the distance between the man and his friend.
To find the distance between the man and his friend, we can use the tangent function:
Tan(angle) = Opposite / Adjacent
In this case, the angle is 16 degrees, the opposite side is x ft, and the adjacent side is 135 ft. Therefore, we can set up the equation as:
Tan(16°) = x / 135
To solve for x, we can multiply both sides by 135:
x = 135 * Tan(16°)
Using a scientific calculator, we can evaluate the right side of the equation:
x ≈ 135 * 0.2865
x ≈ 38.69 ft
So the man is approximately 38.69 feet away from his friend.
Now, let's calculate the distance between the friend and the base of the cliff. We can use the sine function:
Sin(angle) = Opposite / Hypotenuse
In this case, the angle is 16 degrees, the opposite side is 135 ft, and we want to find the hypotenuse, which is the distance between the friend and the base of the cliff.
Sin(16°) = 135 / Hypotenuse
To solve for the hypotenuse, we can rearrange the equation:
Hypotenuse = 135 / Sin(16°)
Using a scientific calculator, we can evaluate the right side of the equation:
Hypotenuse ≈ 135 / 0.2756
Hypotenuse ≈ 489.94 ft
So the friend is approximately 489.94 feet away from the base of the cliff.