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A 26 - f ladder is standing up against a wall. The distance between the base of the ladder and the wall is 14 ft less than the distance between the top of the
ladder and the base of the wall. Find the distance between the base of the ladder and the wall.
The distance between the base of the ladder and the wall is
ft.

Question

UpStudy AI · Accepted Answer

To determine the distance between the base of the ladder and the wall, let's break down the problem step by step.

### **Given:**
- **Length of the ladder (hypotenuse)**: 26 ft
- **Relationship between distances**:
  - Let **$$ x $$** be the horizontal distance from the base of the ladder to the wall.
  - Let **$$ y $$** be the vertical distance from the top of the ladder to the base of the wall.
  
  According to the problem:
  $$
  x = y - 14 \text{ ft}
  $$

### **Using the Pythagorean Theorem:**
Since the ladder forms a right triangle with the wall and the ground:
$$
x^2 + y^2 = 26^2
$$
$$
x^2 + y^2 = 676
$$

### **Substitute $$ x $$ with $$ y - 14 $$:**
$$
(y - 14)^2 + y^2 = 676
$$
$$
y^2 - 28y + 196 + y^2 = 676
$$
$$
2y^2 - 28y + 196 = 676
$$
$$
2y^2 - 28y - 480 = 0
$$

### **Simplify the Equation:**
Divide the entire equation by 2:
$$
y^2 - 14y - 240 = 0
$$

### **Solve the Quadratic Equation:**
Using the quadratic formula $$ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$:
$$
a = 1, \quad b = -14, \quad c = -240
$$
$$
y = \frac{14 \pm \sqrt{(-14)^2 - 4(1)(-240)}}{2}
$$
$$
y = \frac{14 \pm \sqrt{196 + 960}}{2}
$$
$$
y = \frac{14 \pm \sqrt{1156}}{2}
$$
$$
y = \frac{14 \pm 34}{2}
$$

**Possible Solutions:**
1. $$ y = \frac{14 + 34}{2} = 24 \text{ ft} $$
2. $$ y = \frac{14 - 34}{2} = -10 \text{ ft} $$ (Discarded, as distance cannot be negative)

So, $$ y = 24 $$ ft.

### **Find $$ x $$:**
$$
x = y - 14 = 24 - 14 = 10 \text{ ft}
$$

### **Conclusion:**
The distance between the base of the ladder and the wall is **10 feet**.

**Answer:**  
The distance between the base of the ladder and the wall is 10 ft.
The distance between the base of the ladder and the wall is 10 ft.

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