Question

How far is it from the lighthouse to the ship horizontally if the lighthouse is 20 ft tall and shines a beam to the ship 25 ft long?

Also, which statement is correct?



(1 point)
Responses

x = approximately 24.5 yards; from the top of the lighthouse to the boat must be the shortest distance
x = approximately 24.5 yards; from the top of the lighthouse to the boat must be the shortest distance

x = 225 feet; the distance from the lighthouse to the ship must be the longest distance
x = 225 feet; the distance from the lighthouse to the ship must be the longest distance

x = 15 ft; the distance from the lighthouse to the ship must be the shortest distance
x = 15 ft; the distance from the lighthouse to the ship must be the shortest distance

x = 15 ft; the distance from the top of the lighthouse to the ship must be the longest distance

Answer 1

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:

1. **x = approximately 24.5 yards; from the top of the lighthouse to the boat must be the shortest distance** - Incorrect.
2. **x = 225 feet; the distance from the lighthouse to the ship must be the longest distance** - Incorrect.
3. **x = 15 ft; the distance from the lighthouse to the ship must be the shortest distance** - Correct; this represents the shortest horizontal distance.
4. **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.**