Question

A 13 foot ladder is leaning against a wall. The distance from the top of the ladder to the bottom of wall is 7 ft more than the distance from the bottom of the ladder to the wall. Find the distance from the bottom of the ladder to the wall.

Answer 1

Using the Pythagorean theorem, the distance from the bottom of the ladder to the wall is found to be 5 ft, after solving the quadratic equation that arises from the conditions given.

Step-by-step explanation:

The question asks for the distance from the bottom of the ladder to the wall when a ladder is leaning against it. We can use the Pythagorean theorem to solve this problem as it forms a right-angled triangle. Let's denote the distance from the bottom of the ladder to the wall as x, then the distance from the top of the ladder to the bottom of the wall would be x + 7 ft. Since the ladder's length, which represents the hypotenuse, is 13 ft, we can set up the equation:

x2 + (x + 7)2 = 132

Expanding this, we get:

x2 + x2 + 14x + 49 = 169

Combining like terms:

2x2 + 14x - 120 = 0

Dividing everything by 2 to simplify:

x2 + 7x - 60 = 0

Factoring the quadratic equation:

(x + 12)(x - 5) = 0

The possible values for x are -12 and 5. Since distance cannot be negative, the distance from the bottom of the ladder to the wall is 5 ft.

Answer 2

Step-by-step explanation:

The ladder forms a right angle triangle with the wall and the ground. The height of the ladder represents the hypotenuse of the right angle triangle.

Let x represent the distance from the top of the ladder to the bottom of wall. This forms the opposite side of the triangle.

The distance from the top of the ladder to the bottom of wall is 7ft more than the distance from the bottom of the ladder to the wall. This means that the distance from the bottom of the ladder to the wall is x - 7. It represents the adjacent side of the triangle.

To find the distance from the bottom of the ladder to the wall, we would apply Pythagorean theorem.

Hypotenuse² = opposite² + adjacent²

13² = x² + (x - 7)²

169 = x² + x² - 14x + 49

169 = 2x² - 14x + 49

2x² - 14x + 49 - 169 = 0

2x² - 14x - 120 = 0

Dividing through by 2, it becomes

x² - 7x - 60 = 0

x² + 5x - 12x - 60 = 0

x(x + 5) - 12(x + 5) = 0

(x - 12)(x + 5) = 0

x = 12 or x = - 5

the distance cannot be negative, so x = 12

the distance from the bottom of the ladder to the wall is

x - 7 = 12 - 7 = 5 feet