Question
A picture is shown with a coastguard, lighthouse, and ship distance with a right triangle being shown
The distance between the coastguard and lighthouse is 400 yards while the distance between the coastguard and ship is 500 yards, with the distance between the lighthouse and ship being x
How far from the lighthouse to the ship?
A correct statement: A. The distance from, the coastguard to ship must be the shortest distance. B. From the lighthouse to the coastguard must be the shortest distance. C. From coastguard to lighthouse must be the longest distance. D. The distance from the coastguard to the ship must be the longest distance
The distance between the coastguard and lighthouse is 400 yards while the distance between the coastguard and ship is 500 yards, with the distance between the lighthouse and ship being x
How far from the lighthouse to the ship?
A correct statement: A. The distance from, the coastguard to ship must be the shortest distance. B. From the lighthouse to the coastguard must be the shortest distance. C. From coastguard to lighthouse must be the longest distance. D. The distance from the coastguard to the ship must be the longest distance
Answers
GPT-4o mini
To solve the problem, we can use the properties of a right triangle, given the distances involved.
We have:
- Distance from the coastguard to the lighthouse (C to L): 400 yards
- Distance from the coastguard to the ship (C to S): 500 yards
- Distance from the lighthouse to the ship (L to S): x
According to the Pythagorean theorem, for a right triangle where:
- One leg is the distance between the coastguard and the lighthouse (400 yards),
- The other leg is the distance from the coastguard to the ship (500 yards),
- The hypotenuse would be the distance from the lighthouse to the ship (x).
We can set up the equation:
\[
x^2 = (C to L)^2 + (C to S)^2
\]
Substituting the known values:
\[
x^2 = 400^2 + 500^2
\]
\[
x^2 = 160000 + 250000
\]
\[
x^2 = 410000
\]
Taking the square root of both sides gives us:
\[
x = \sqrt{410000} \approx 640.31 \text{ yards}
\]
Now let's evaluate the statements given:
A. The distance from the coastguard to ship must be the shortest distance. (Incorrect; it's actually the longest distance.)
B. From the lighthouse to the coastguard must be the shortest distance. (Incorrect; while it's shorter than the distance to the ship, it's not the shortest overall because of the distance to the ship.)
C. From coastguard to lighthouse must be the longest distance. (Incorrect; it's not the longest distance; the distance from coastguard to ship is longer.)
D. The distance from the coastguard to the ship must be the longest distance. (Correct; it is longer than both the distance to the coastguard and the distance to the lighthouse.)
The correct answer is:
**D. The distance from the coastguard to the ship must be the longest distance.**
We have:
- Distance from the coastguard to the lighthouse (C to L): 400 yards
- Distance from the coastguard to the ship (C to S): 500 yards
- Distance from the lighthouse to the ship (L to S): x
According to the Pythagorean theorem, for a right triangle where:
- One leg is the distance between the coastguard and the lighthouse (400 yards),
- The other leg is the distance from the coastguard to the ship (500 yards),
- The hypotenuse would be the distance from the lighthouse to the ship (x).
We can set up the equation:
\[
x^2 = (C to L)^2 + (C to S)^2
\]
Substituting the known values:
\[
x^2 = 400^2 + 500^2
\]
\[
x^2 = 160000 + 250000
\]
\[
x^2 = 410000
\]
Taking the square root of both sides gives us:
\[
x = \sqrt{410000} \approx 640.31 \text{ yards}
\]
Now let's evaluate the statements given:
A. The distance from the coastguard to ship must be the shortest distance. (Incorrect; it's actually the longest distance.)
B. From the lighthouse to the coastguard must be the shortest distance. (Incorrect; while it's shorter than the distance to the ship, it's not the shortest overall because of the distance to the ship.)
C. From coastguard to lighthouse must be the longest distance. (Incorrect; it's not the longest distance; the distance from coastguard to ship is longer.)
D. The distance from the coastguard to the ship must be the longest distance. (Correct; it is longer than both the distance to the coastguard and the distance to the lighthouse.)
The correct answer is:
**D. The distance from the coastguard to the ship must be the longest distance.**