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A diagram illustrating a problem, likely in geometry or trigonometry. 

The scene displays a nautical setting, with a Coast Guard station and a lighthouse situated along a horizontal line. A ship is shown at a certain distance from both. 

Measurements are labeled along the horizontal line: 400 yards between the Coast Guard and lighthouse; and a diagonal measurement of 900 yards from the ship to the Coast Guard.  

A line extends from the ship to the Coast Guard, and another line extends from the ship to the lighthouse (forming a triangle). 

The triangle illustrates a scenario likely requiring calculation of distance or angles. The problem involves finding the unknown length or angle marked by a variable (possibly for distance or angle to the ship from the lighthouse).

A small text box on the right of the diagram reads "(2 points)". This likely indicates the expectation to solve the geometry problem and provide two key pieces of information or calculations.

How far from the lighthouse to the ship? (Response Area)

A correct statement (Response Area)

Answers:
The distance from the coastguard to ship must be the longest distance
from the lighthouse to the coastguard must be the shortest distance
20 yds
300 yds
the distance from the coastguard to the ship must be the shortest distance
24.5 yds
490 ft
from the coastguard to the lighthouse must be the longest distance

1 answer

To solve the problem described in the nautical setting involving the Coast Guard station, lighthouse, and ship, we can apply the Pythagorean theorem, as we are dealing with a right triangle formed by these points.

The diagram indicates:

  • The distance between the Coast Guard station and the lighthouse is 400 yards (horizontal distance).
  • The distance from the ship to the Coast Guard station is 900 yards (this represents the hypotenuse of the triangle).
  • We need to find the distance from the ship to the lighthouse, which we can represent as "d."

Using the Pythagorean theorem, we have:

\[ a^2 + b^2 = c^2 \]

Where:

  • \( a \) = distance from the Coast Guard station to the lighthouse (400 yards)
  • \( b \) = distance from the ship to the lighthouse (d)
  • \( c \) = distance from the ship to the Coast Guard station (900 yards)

In this case: \[ (400)^2 + d^2 = (900)^2 \]

Calculating: \[ 160000 + d^2 = 810000 \]

Subtracting 160000 from both sides: \[ d^2 = 810000 - 160000 \] \[ d^2 = 650000 \]

Now taking the square root of both sides: \[ d = \sqrt{650000} \approx 806.23 \text{ yards} \]

Conclusion:

The distance from the lighthouse to the ship is approximately 806.23 yards.

Regarding the correct statement from the provided options:

  • The distance from the Coast Guard to the ship must be the longest distance. (This is correct because 900 yards is indeed the longest distance in the triangle formed.)

Thus, the correct responses are:

  • The distance from the lighthouse to the ship is approximately 806.23 yards.
  • The distance from the Coast Guard to the ship must be the longest distance.