Question

PART X: Lost at sea

Suddenly, during your race, a powerful thunderstorm strikes. A wave washes over your boat, shorting out your navigation devices. They wouldn't be much use anyhow, because the lightning is interfering with all communications in the area.
When the storm ends and the sea is calm, you're safe, because you observed all safety precautions, but you have no idea how far off course you are. However, you do know:
* The height of a nearby landmark, which you remember from the tourist center you visited yesterday.
> Mykonos: The light in the Armenistis Lighthouse is 604 feet above sea level.
* The length of your thumb - it's about 2 inches long.
* That you can estimate lengths of less than a foot pretty accurately (to the nearest inch).

1. First, you close one eye and hold your thumb up to block your view of the landmark. You move your thumb nearer and farther from your eye until it just covers the landmark.

Make a sketch showing overlapping triangles ΔEBT and ΔELH to illustrate this situation, where your eye is E, the base of your thumb is B, the tip of your thumb is T, the bottom of the landmark is L, and the top of the landmark is H.

2. Explain why ΔEBT~ΔELH

3. When your thumb covers the landmark, you estimate that it is 10 inches from your eye. Label your drawing with the known measurements.

4. Solve for EL, the distance from your eye to the landmark. Show all work and justify each step.

Answer 1

Answer: 1. Here is a sketch of the overlapping triangles ΔEBT and ΔELH:

H

/|

/ |

/ |

/ |

/ θ |

/____|L

E

|

|

|

|

T

|

|

|

|

B

2. ΔEBT~ΔELH because they have the same shape (both are right triangles with angle θ in common) and their corresponding sides are proportional. Specifically, angle θ is shared by both triangles, angle EBT is a right angle, and angle ELH is a right angle. Therefore, by the Angle-Angle Similarity Theorem, the two triangles are similar. Corresponding sides are proportional because EB/EL = BT/LH (by definition of similar triangles).

3. We are given that BT (the length of the thumb) is about 2 inches. We are also told that the distance from the eye to the landmark (EL) is unknown, but that the distance from the eye to the thumb (EB) is 10 inches (estimated by the person). We are given that LH (the height of the landmark) is 604 feet, or 604*12=7248 inches (since there are 12 inches in a foot). Therefore, we can label the diagram as follows:

7248

/|

/ |

/ | EL

/ |

/ θ |

/____|L

E

|10

|

|

|

T

| 2

|

|

|

|

B

4. To solve for EL, we use the fact that the triangles are similar: EB/EL = BT/LH. Substituting in the known values, we get:

10/EL = 2/7248

5. Multiplying both sides by EL and dividing both sides by 2, we get:

EL = 7248*5 = 36240 inches

Therefore, the distance from the eye to the landmark is 36240/12 = 3020 feet (rounded to the nearest foot).

Explanation: