To find the horizontal distance from the lighthouse to the ship, we can use the Pythagorean theorem. We have a right triangle where:
- One leg (height of the lighthouse) = 20 ft
- The other leg (horizontal distance) = x (unknown)
- Hypotenuse (length of the beam) = 25 ft
According to the Pythagorean theorem:
\[ a^2 + b^2 = c^2 \]
Substituting in the values we know:
\[ 20^2 + x^2 = 25^2 \]
Calculating the squares:
\[ 400 + x^2 = 625 \]
Now, subtract 400 from both sides:
\[ x^2 = 625 - 400 \] \[ x^2 = 225 \]
Taking the square root of both sides:
\[ x = \sqrt{225} = 15 \text{ ft} \]
So the horizontal distance from the lighthouse to the ship is 15 ft.
Now, let's evaluate the statements given:
-
x = approximately 24.5 yards; from the top of the lighthouse to the boat must be the shortest distance - This is incorrect.
-
x = 15 ft; the distance from the top of the lighthouse to the ship must be the longest distance - This is correct; the distance from the top of the lighthouse (the hypotenuse) to the ship is longer than the horizontal distance (15 ft).
-
x = 15 ft; the distance from the lighthouse to the ship must be the shortest distance - This is incorrect in the context of comparing distances (the beam's length is longer).
-
x = 225 feet; the distance from the lighthouse to the ship must be the longest distance - This is incorrect; the distance is 15 ft, not 225 ft.
Correct statement:
- x = 15 ft; the distance from the top of the lighthouse to the ship must be the longest distance (second option).