A lighthouse that rises 49 ft above the surface of the water sits on a rocky cliff that extends 19 ft from its base. A sailor on the deck of a ship sights the top of the lighthouse at an angle of 30.0 ∘ above the horizontal. If the sailor's eye level is 14 ft above the water, how far is the ship from the rocks?

I attempted to find the hypotenus of the triangle that I drew and I kept getting a negative number. Obviously that is not right. Thanks for the help.

Tan30 = (49-14)/(X1+X2)

Tan30 = 35/(19+X2)
(19+X2)Tan30 = 35
(19+X2) = 35/Tan30 = 60.62
X2=60.62-19 = 41.62 Ft. from the rocks.

Ah, the sailor is in quite a predicament, isn't he? Let's help him out! We'll use some trigonometry to figure this out.

First, let's draw a right triangle with the lighthouse, the sailor's eye level, and the ship's distance from the rocks. The height of the lighthouse above the surface of the water is 49 ft, and the sailor's eye level is 14 ft above the water. The distance from the base of the lighthouse to the rocks is 19 ft.

Now, we know that the angle at which the sailor sights the top of the lighthouse is 30 degrees above the horizontal. That means the angle between the sailor's eye level and the line connecting the sailor to the top of the lighthouse is 60 degrees (since the sum of the angles in a triangle is 180 degrees).

Now, let's use some trigonometry magic! The tangent of 60 degrees is equal to the height of the lighthouse (49 ft) divided by the distance from the ship to the rocks (let's call it x). So, we have:

tan(60 degrees) = 49 ft / x

To solve for x, we can rearrange the equation:

x = 49 ft / tan(60 degrees)

Now, let's calculate that:

x ≈ 49 ft / 1.732

x ≈ 28.28 ft

So, the ship is approximately 28.28 ft away from the rocks.

Hope that helps, and happy sailing!

To solve this problem, we can use trigonometry and the concept of similar triangles.

Let's label the points in the problem. The base of the lighthouse is point A, the top of the lighthouse is point B, the sailor's eye level is point C, and the ship's position is point D. The distance from the ship to the rocks is x.

We can create two right triangles: ABC and ADC.

In triangle ABC, we have:

- The height of the lighthouse, AB = 49 ft.
- The distance from the cliff to the lighthouse, AC = 19 ft.
- The angle above the horizontal from the sailor to the top of the lighthouse, ∠BAC = 30°.

In triangle ADC, we have:

- The height of the sailor's eye level, CD = 14 ft.
- The distance from the ship to the rocks, AD = x.

Our objective is to determine the value of x.

Now, let's begin solving the problem step by step:

Step 1: Calculate the lengths of sides BC and AC.

Using trigonometry, we know that:

BC = AB * tan(∠BAC)

BC = 49 ft * tan(30°)

BC ≈ 28.34 ft

AC = AB - CD

AC = 49 ft - 14 ft

AC = 35 ft

Step 2: Set up a proportion between triangles ABC and ADC.

Since triangles ABC and ADC are similar, we can set up a proportion between their corresponding sides:

BC / AC = AD / CD

Substituting the values we found:

28.34 ft / 35 ft = x / 14 ft

Step 3: Solve the proportion for x.

Cross multiplying the proportion:

28.34 ft * 14 ft = 35 ft * x

397.16 ft = 35 ft * x

Divide both sides by 35 ft:

397.16 ft / 35 ft = x

x ≈ 11.34 ft

Therefore, the ship is approximately 11.34 ft away from the rocks.

To solve this problem, we can use the trigonometric functions sine, cosine, and tangent, along with the given information about angles and distances.

First, let's draw a diagram to visualize the situation. The lighthouse is located on a rocky cliff, and the sailor on the ship sees the top of the lighthouse at an angle of 30 degrees above the horizontal.

```
C
/|
/ |
/ |49 ft
/θ |
/ |
/ |
ship------B ----19 ft--A (lighthouse base)
\ |
\ |
\ |
\ |
\|
eye
```

Here are the steps to solve the problem:

1. Find the height of the lighthouse above the eye level of the sailor on the ship.
- The total height of the lighthouse is 49 ft, and the sailor's eye level is 14 ft above the water. Therefore, the height of the lighthouse above the sailor's eye level is 49 ft - 14 ft = 35 ft.

2. Find the distance between the ship (point B) and the base of the lighthouse (point A).
- The rocky cliff extends 19 ft from the base of the lighthouse. Therefore, the distance between points B and A is 19 ft.

3. Use trigonometry to find the distance between the ship and the rocks (point C).
In the right triangle ABC, we can use the tangent function (tan θ) to calculate the ratio of the opposite side (35 ft) to the adjacent side (distance BC) of angle θ:

tan θ = opposite / adjacent
tan θ = 35 ft / BC

Rearranging the equation to solve for BC:
BC = 35 ft / tan θ

Using a calculator, calculate the value of tan 30 degrees:
tan 30° ≈ 0.5774

Plug in the values:
BC = 35 ft / 0.5774

BC ≈ 60.64 ft

So, the ship is approximately 60.64 ft away from the rocks.

Keep in mind that negative values might arise if you mistakenly use the reverse ratios (e.g., cosine or sine) or if you input the angle in degrees instead of radians when using trigonometric functions.