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. 20 yards. B. 24.5 yards. C. 300 yards. D. 490 yards
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. 20 yards. B. 24.5 yards. C. 300 yards. D. 490 yards
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 for the distance between the lighthouse and the ship (denoted as \( x \)), we can use the Pythagorean theorem, which applies since the coastguard, lighthouse, and ship form a right triangle.
From the information provided:
- The distance between the coastguard and the lighthouse (one leg of the triangle) is 400 yards.
- The distance between the coastguard and the ship (the hypotenuse of the triangle) is 500 yards.
- The distance between the lighthouse and the ship (the other leg of the triangle) is \( x \).
According to the Pythagorean theorem:
\[
AC^2 + BC^2 = AB^2
\]
where
- \( AC \) is the distance from the coastguard to the lighthouse (400 yards),
- \( BC \) is the distance from the lighthouse to the ship (\( x \)),
- \( AB \) is the distance from the coastguard to the ship (500 yards).
Substituting in the values, the equation becomes:
\[
400^2 + x^2 = 500^2
\]
Calculating the squares:
\[
160000 + x^2 = 250000
\]
Now, isolate \( x^2 \):
\[
x^2 = 250000 - 160000
\]
\[
x^2 = 90000
\]
Taking the square root of both sides:
\[
x = \sqrt{90000} = 300 \text{ yards}
\]
Thus, the distance from the lighthouse to the ship is **300 yards**.
Regarding the statements provided:
A. The distance from the coastguard to the ship must be the shortest distance. **(Incorrect)**
B. From the lighthouse to the coastguard must be the shortest distance. **(Incorrect)**
C. From coastguard to lighthouse must be the longest distance. **(Incorrect)**
D. The distance from the coastguard to the ship must be the longest distance. **(Correct)**
The correct answer is **C: 300 yards** and statement **D** is the only correct statement.
From the information provided:
- The distance between the coastguard and the lighthouse (one leg of the triangle) is 400 yards.
- The distance between the coastguard and the ship (the hypotenuse of the triangle) is 500 yards.
- The distance between the lighthouse and the ship (the other leg of the triangle) is \( x \).
According to the Pythagorean theorem:
\[
AC^2 + BC^2 = AB^2
\]
where
- \( AC \) is the distance from the coastguard to the lighthouse (400 yards),
- \( BC \) is the distance from the lighthouse to the ship (\( x \)),
- \( AB \) is the distance from the coastguard to the ship (500 yards).
Substituting in the values, the equation becomes:
\[
400^2 + x^2 = 500^2
\]
Calculating the squares:
\[
160000 + x^2 = 250000
\]
Now, isolate \( x^2 \):
\[
x^2 = 250000 - 160000
\]
\[
x^2 = 90000
\]
Taking the square root of both sides:
\[
x = \sqrt{90000} = 300 \text{ yards}
\]
Thus, the distance from the lighthouse to the ship is **300 yards**.
Regarding the statements provided:
A. The distance from the coastguard to the ship must be the shortest distance. **(Incorrect)**
B. From the lighthouse to the coastguard must be the shortest distance. **(Incorrect)**
C. From coastguard to lighthouse must be the longest distance. **(Incorrect)**
D. The distance from the coastguard to the ship must be the longest distance. **(Correct)**
The correct answer is **C: 300 yards** and statement **D** is the only correct statement.