A ladder is 1 foot longer than its vertical heigt along the side of the house. The distance of the base of the ladder from the house is 7 feet less than its vertical height from the ground. How far up the house does the ladder reach?
To find out how far up the house the ladder reaches, we need to break down the given information and solve for the height.
Let's assume the vertical height along the side of the house is represented by 'h' feet.
According to the problem, the ladder is 1 foot longer than its vertical height, which means the length of the ladder can be represented as 'h + 1' feet.
The distance of the base of the ladder from the house is 7 feet less than its vertical height from the ground. So, the distance of the base can be represented as 'h - 7' feet.
Now, we have a right triangle formed by the ladder, the height of the house, and the distance of the base from the house. The ladder acts as the hypotenuse of the right triangle.
We can use the Pythagorean theorem, which states that in any right triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.
So, applying the Pythagorean theorem, we have:
(h + 1)^2 = h^2 + (h - 7)^2
Expanding and simplifying the equation:
h^2 + 2h + 1 = h^2 + h^2 - 14h + 49
Combine like terms:
2h + 1 = 2h^2 - 14h + 49
Rearranging the equation:
2h^2 - 16h + 48 = 0
Dividing the equation by 2 to simplify:
h^2 - 8h + 24 = 0
Now, we can solve this quadratic equation using factoring, completing the square, or the quadratic formula. Factoring the equation, we have:
(h - 4)(h - 6) = 0
So, h - 4 = 0 or h - 6 = 0
Solving for 'h' in each equation:
h = 4 or h = 6
Since we are looking for the height of the ladder, we can only consider the positive value, h = 6.
Therefore, the ladder reaches a height of 6 feet up the house.