The foot of an extension ladder is 9ft from a wall. The height that the ladder reaches on the wall and the length of the ladder are consecutive integers. How long is the ladder?
The simplest ratio of sides of a right-angled triangle is the 3:4:5 ratio
so we have 9:x:y, clearly x = 12, and y = 15
check: 9^2 + 12^2 = 15^2 , YES
9^2+x^2=(x+1)^2
81+x^2 = x^2+2x+1
x=40
So, the ladder is 41 ft long.
That's some ladder!
Let's assume that the height the ladder reaches on the wall is x, and the length of the ladder is x + 1 (since they are consecutive integers).
According to the Pythagorean theorem, the square of the hypotenuse (the ladder) is equal to the sum of the squares of the other two sides. In this case, the hypotenuse is the ladder, the other two sides are the distance from the foot of the ladder to the wall (9 ft) and the height the ladder reaches on the wall (x).
Using the Pythagorean theorem, we can write the equation as follows:
(9)^2 + (x)^2 = (x + 1)^2
Expanding the equation:
81 + x^2 = x^2 + 2x + 1
Simplifying the equation:
81 = 2x + 1
2x = 80
Dividing both sides by 2:
x = 40
So, the height that the ladder reaches on the wall is 40 ft, and the length of the ladder is 40 + 1 = 41 ft.
Therefore, the ladder is 41 ft long.
To solve this problem, we can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.
Let's assume that the height the ladder reaches on the wall is x, and the length of the ladder is x + 1 (since they are consecutive integers).
According to the problem, the foot of the ladder is 9 feet away from the wall. So, we can consider the foot, the wall, and the ladder as the three sides of a right triangle. The distance between the foot of the ladder and the height it reaches is the base of the triangle, the height it reaches is the height of the triangle, and the ladder itself is the hypotenuse.
Using the Pythagorean theorem, we have:
(9)^2 + (x)^2 = (x + 1)^2
Simplifying this equation, we get:
81 + x^2 = x^2 + 2x + 1
Rearranging the equation, we get:
2x = 80
Dividing both sides by 2, we have:
x = 40
Therefore, the height the ladder reaches on the wall is 40 feet, and the length of the ladder is 40 + 1 = 41 feet. Hence, the ladder is 41 feet long.