I need to find the slope of a ladder that is leaning against a building. The bottom of the ladder is 9 ft from the building and the top of the ladder is 16 ft above the ground.
slope = rise/run = 16/9
To find the slope of the ladder, you need to calculate the ratio of the vertical height to the horizontal distance. In this case, the vertical height is the length of the ladder, and the horizontal distance is the distance from the bottom of the ladder to the building.
The length of the ladder can be found using the Pythagorean theorem, which states that the square of the hypotenuse (the ladder) is equal to the sum of the squares of the other two sides (the vertical height and the horizontal distance).
Let's calculate the length of the ladder first:
Length of ladder^2 = (vertical height)^2 + (horizontal distance)^2
(vertical height)^2 = 16^2 = 256
(horizontal distance)^2 = 9^2 = 81
Length of ladder^2 = 256 + 81 = 337
Taking the square root of both sides:
Length of ladder = square root of 337 ≈ 18.357 ft
Now that we have the length of the ladder and the horizontal distance, we can find the slope:
Slope = vertical height / horizontal distance = 16 ft / 9 ft ≈ 1.778
Therefore, the slope of the ladder is approximately 1.778.
To find the slope of the ladder, we can use the concept of right triangles. The ratio of the length of the ladder to the distance of the bottom of the ladder from the building is equal to the ratio of the height of the ladder to the distance from the top of the ladder to the ground.
In this case, the length of the ladder (hypotenuse) is the distance from the top of the ladder to the ground, which is 16 ft. The distance from the bottom of the ladder to the building is 9 ft.
So, let's label the length of the ladder as 'L', the height of the ladder as 'H', and the distance from the top of the ladder to the ground as 'D'.
We have the following relationship using the Pythagorean theorem:
L^2 = H^2 + D^2
Given that H = 16 ft and D = 9 ft, we can substitute these values into the equation:
L^2 = 16^2 + 9^2
L^2 = 256 + 81
L^2 = 337
To find L, we take the square root of both sides of the equation: √L^2 = √337.
Therefore, the length of the ladder (L) is approximately √337 ft.
To find the slope of the ladder, we can use the formula:
Slope = H / D
Substituting the given values, we get:
Slope = 16 ft / 9 ft
Therefore, the slope of the ladder leaning against the building is approximately 1.778 ft.