A painter leans a 15 foot ladder against the side of a house. The top of the ladder touches the house at a height that is 3 feet more than the distance from the house to the bottom of the ladder. How far is the bottom of the ladder from the house?
x^2 + (x+3)^2 = 15^2
hint: think of a 3-4-5 right triangle
15*15=225
x*x=x^2
(x+3)*(x+3)=X^2+9
225=x^2+x^2+9
225=2x^2+9
216=2x^2
108=x^2
10.39=x
To find the distance from the house to the bottom of the ladder, we can use a basic algebraic equation.
Let's assume the distance from the bottom of the ladder to the house is 'x' feet.
According to the problem, the top of the ladder touches the house at a height that is 3 feet more than the distance from the house to the bottom of the ladder. So, the height up the house is 'x + 3' feet.
The ladder forms a right triangle with the house and the ground, where the ladder is the hypotenuse. Using the Pythagorean theorem, we can set up the equation:
x^2 + (x + 3)^2 = 15^2
Expanding the equation:
x^2 + (x^2 + 6x + 9) = 225
Combining like terms:
2x^2 + 6x + 9 = 225
Now, let's solve for 'x' by bringing all terms to one side to form a quadratic equation:
2x^2 + 6x + 9 - 225 = 0
2x^2 + 6x - 216 = 0
To simplify the equation, let's divide all terms by 2:
x^2 + 3x - 108 = 0
Now, we can solve this quadratic equation by factoring or using the quadratic formula.
Factoring this equation might be complicated, so let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a = 1, b = 3, and c = -108. Substituting these values into the quadratic formula:
x = (-3 ± √(3^2 - 4 * 1 * -108)) / (2 * 1)
Simplifying this equation:
x = (-3 ± √(9 + 432)) / 2
x = (-3 ± √(441)) / 2
x = (-3 ± 21) / 2
Now we have two possible solutions:
x₁ = (-3 + 21)/2
x₁ = 18/2
x₁ = 9
x₂ = (-3 - 21)/2
x₂ = -24/2
x₂ = -12
Since the distance cannot be negative, we reject x₂ = -12.
Therefore, the distance from the bottom of the ladder to the house is 9 feet.