3) A 14-foot-long ladder is leaning against a building. The height from the ground to where the ladder touches the building is 5 feet more than three times the distance from the base of the building to the bottom of the ladder. What is the distance from building's base to the top of ladder and the distance from the building's base to the bottom of the ladder?

Let's denote the distance from the base of the building to the bottom of the ladder as x feet. According to the problem, the height from the ground to where the ladder touches the building is 5 feet more than three times the distance from the base of the building to the bottom of the ladder. So, the height can be represented as 3x + 5 feet.<br /><br />We can use the Pythagorean theorem to solve this problem. The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.<br /><br />In this case, the ladder acts as the hypotenuse, and the height and the distance from the building's base to the bottom of the ladder are the other two sides. So, we can write the equation as:<br /><br />(14)^2 = (3x + 5)^2 + x^2<br /><br />Solving this equation, we get:<br /><br />196 = 9x^2 + 30x + 25 + x^2<br />196 = 10x^2 + 30x + 25<br />0 = 10x^2 + 30x + 25 - 196<br />0 = 10x^2 + 30x - 171<br /><br />Using the quadratic formula, we can solve for x:<br /><br />x = (-30 ± √(30^2 - 4 * 10 * (-171))) / (2 * 10)<br />x = (-30 ± √(900 + 6840)) / 20<br />x = (-30 ± √(7740)) / 20<br />x = (-30 ± 87.7) / 20<br /><br />We have two possible solutions for x:<br />x = (-30 + 87.7) / 20 ≈ 2.885<br />x = (-30 - 87.7) / 20 ≈ -5.885<br /><br />Since the distance cannot be negative, we take the positive value:<br />x ≈ 2.885 feet<br /><br />Now, we can find the height by substituting x back into the equation for the height:<br />Height = 3x + 5<br />Height ≈ 3 * 2.885 + 5<br />Height ≈ 8.755 + 5<br />Height ≈ 13.755 feet<br /><br />Therefore, the distance from the building's base to the bottom of the ladder is approximately 2.885 feet, and the distance from the building's base to the top of the ladder is approximately 13.755 feet.