A 10 foot ladder is placed against a vertical wall of a building, with the bottom of the ladder standing on level ground 8 feet from the base of the building. How high up the wall does the ladder reach.
A 11-foot ladder is placed against a vertical wall of a building, with the bottom of the ladder standing on level ground 8 feet from the base of the building. How high up the wall does the ladder reach?
answer is 6
Well, if a 10-foot ladder is placed against a vertical wall, it's pretty impressive that it reached anywhere at all – those walls can be quite stubborn. But let's put my skepticism aside for a moment and solve this mathematically.
So, according to the problem, the ladder forms a right-angled triangle with the wall and the ground. The length of the ladder itself represents the hypotenuse of this triangle. We can use the Pythagorean theorem to find the height, which states that the sum of the squares of the two shorter sides is equal to the square of the hypotenuse.
So, let's do the math: 8 squared plus the height squared equals 10 squared. That simplifies to 64 + height squared equals 100. If we subtract 64 from both sides, we get height squared equals 36. Finally, taking the square root of both sides, the height is equal to 6 feet.
Therefore, this ladder reaches a height of 6 feet up the wall! Even walls can't resist the charm of a ladder.
To find out how high up the wall the ladder reaches, we can use the Pythagorean theorem, which states that in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the two other sides.
In this case, the ladder is the hypotenuse of the triangle, and the distance from the base of the building to the ladder is one of the other sides. We'll call this distance "a."
Step 1: Identify the values
- The distance from the base of the building to the ladder (a) = 8 feet
- The length of the ladder (the hypotenuse) = 10 feet
Step 2: Apply the Pythagorean theorem
According to the theorem, we have:
a^2 + b^2 = c^2
Plugging in the known values, we get:
8^2 + b^2 = 10^2
Step 3: Solve for b
64 + b^2 = 100
b^2 = 100 - 64
b^2 = 36
Taking the square root of both sides, we find:
b = √36
b = 6
So, the ladder reaches a height of 6 feet up the wall.
To determine how high up the wall the ladder reaches, we can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (the longest side, which is the ladder in this case) is equal to the sum of the squares of the other two sides.
Let's designate the height up the wall that the ladder reaches as "x." The base of the triangle is 8 feet, the height is x feet, and the hypotenuse is 10 feet.
Using the Pythagorean theorem, we can write the equation as:
8^2 + x^2 = 10^2
Simplifying:
64 + x^2 = 100
Subtracting 64 from both sides of the equation:
x^2 = 36
Taking the square root of both sides:
x = √36
Since we're looking for a positive value, x = 6.
Therefore, the ladder reaches a height of 6 feet up the wall.
a^2 + b^2 = c^2
8^2 + b^2 = 10^2
64 + b^2 = 100
b^2 = 36
b = 6
The ladder reaches 8 feet.