To solve the problem, we can use the Pythagorean theorem. The situation can be visualized as a right triangle where:
- The height of the lighthouse (20 ft) acts as one leg.
- The beam of light to the ship (25 ft) acts as the hypotenuse.
- We need to find the horizontal distance (x) from the lighthouse to the ship, which acts as the other leg of the triangle.
According to the Pythagorean theorem:
\[ a^2 + b^2 = c^2 \]
where:
- \(a\) = horizontal distance (x)
- \(b\) = height of the lighthouse (20 ft)
- \(c\) = length of the beam (25 ft)
Now we can set up the equation:
\[ x^2 + 20^2 = 25^2 \]
Calculating the squares:
\[ x^2 + 400 = 625 \]
Now, subtract 400 from both sides:
\[ x^2 = 625 - 400 \] \[ x^2 = 225 \]
Now take 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, to evaluate the statements:
- x = approximately 24.5 yards; from the top of the lighthouse to the boat must be the shortest distance - Incorrect.
- x = 225 feet; the distance from the lighthouse to the ship must be the longest distance - Incorrect.
- x = 15 ft; the distance from the lighthouse to the ship must be the shortest distance - Correct; this represents the shortest horizontal distance.
- x = 15 ft; the distance from the top of the lighthouse to the ship must be the longest distance - Incorrect; the hypotenuse (25 ft) is the longest distance.
Thus, the correct statement is:
x = 15 ft; the distance from the lighthouse to the ship must be the shortest distance.