A ladder 12 ft. long leans against a wall, touching it 8 ft. above the ground. What is the slope of the ladder?
Pythagorean Theorem:
a^2 + b^2 = c^2
8^2 + b^2 = 12^2
64 + b^2 = 144
b^2 = 80
b = 8.9 feet
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Well, the ladder is definitely still pulling some strings even as it leans against that wall! The slope of the ladder can be found by simply dividing the rise (8 ft.) by the run (12 ft.). So, the slope of the ladder would be 8/12, which simplifies to 2/3. So, the ladder is sloping at a ratio of 2 to 3! Keep reaching for those heights!
To find the slope of the ladder, we need to determine the ratio between the rise (vertical distance) and the run (horizontal distance) of the ladder.
In this case, the rise of the ladder is the distance between the ground and the point where the ladder touches the wall, which is 8 ft.
The run of the ladder is the length of the ladder itself, which is 12 ft.
Therefore, the slope of the ladder is determined by the rise over the run:
Slope = Rise/Run = 8/12 = 2/3
So, the slope of the ladder is 2/3.
To find the slope of the ladder, you can use a right triangle formed by the ladder, the wall, and the ground. The length of the ladder is the hypotenuse, the distance from the base of the ladder to the wall is the adjacent side, and the distance from the base of the ladder to the ground is the opposite side.
Using the Pythagorean theorem, we can solve for the length of the base of the triangle (the adjacent side).
The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
In this case, the hypotenuse (the ladder) is 12 ft., and the opposite side (the distance from the base of the ladder to the ground) is 8 ft.
Let's call the length of the adjacent side x:
x^2 + 8^2 = 12^2
x^2 + 64 = 144
x^2 = 144 - 64
x^2 = 80
Taking the square root of both sides:
x = sqrt(80)
Now, we can calculate the slope of the ladder by finding the ratio of the opposite side (8 ft.) to the adjacent side (the length of the base of the triangle):
slope = opposite/adjacent
slope = 8/sqrt(80)
Simplifying,
slope ≈ 0.89 (rounded to two decimal places)
So, the slope of the ladder is approximately 0.89.