A 15-ft ladder leans against a wall. The top touches the wall at a height of 13 ft. How far is the base of the ladder from the wall?

Rationale

The base of the ladder is 2√14 ft from the wall.

Using the Pythagorean theorem, we can calculate the distance from the base of the ladder to the wall. The ladder, wall, and ground form a right triangle, where the ladder is the hypotenuse (15 ft), the height at which it touches the wall is one leg (13 ft), and the distance from the wall to the base of the ladder is the other leg, which we need to find.

A (7 ft) YOUR ANSWER

If the base of the ladder were 7 ft from the wall, we could apply the Pythagorean theorem: (7^2 + 13^2 = 49 + 169 = 218). The hypotenuse squared would be (15^2 = 225), which does not equal 218, indicating that this choice is incorrect.

B (2√14 ft) CORRECT

This choice is correct. Using the Pythagorean theorem, we set up the equation as follows: (x^2 + 13^2 = 15^2). This simplifies to (x^2 + 169 = 225), leading to (x^2 = 56) and therefore (x = √56 = 2√14). This confirms the distance from the wall to the base of the ladder.

C (7.5 ft) YOUR ANSWER

If the base of the ladder were 7.5 ft from the wall, we would have (7.5^2 + 13^2 = 56.25 + 169 = 225.25). Again, the hypotenuse squared is (15^2 = 225), showing that this choice does not satisfy the Pythagorean theorem, making it incorrect.

D (8 ft) YOUR ANSWER

If the base were 8 ft from the wall, we would calculate (8^2 + 13^2 = 64 + 169 = 233). The hypotenuse squared is (15^2 = 225), which does not match, confirming that this option is incorrect as well.

Conclusion

The distance from the base of the ladder to the wall is accurately calculated using the Pythagorean theorem, leading us to the conclusion that it is (2√14) ft. This value is derived from the relationship between the sides of a right triangle formed by the ladder, the wall, and the ground, illustrating the fundamental principles of geometry in practical application.

A 15-ft ladder leans against a wall. The top touches the wall at a height of 13 ft. How far is the base of the ladder from the wall?

The base of the ladder is 2√14 ft from the wall.

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