A 13
-foot
ladder is placed against a vertical wall of a building, with the bottom of the ladder standing on level ground 6
feet from the base of the building. How high up the wall does the ladder reach?
We can use the Pythagorean theorem to solve this problem. Let's label the height up the wall that the ladder reaches as "x". Then we have:
a = 6 feet (the distance from the base of the building to the bottom of the ladder)
b = x feet (the height up the wall that the ladder reaches)
c = 13 feet (the length of the ladder)
According to the Pythagorean theorem, a^2 + b^2 = c^2. Substituting in our values gives us:
6^2 + x^2 = 13^2
36 + x^2 = 169
x^2 = 133
x ≈ 11.54
So the ladder reaches a height of approximately 11.54 feet up the wall.
To determine how high up the wall the ladder reaches, we can use the Pythagorean theorem. The theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
In this problem, the ladder represents the hypotenuse, and the distance from the base of the building to the wall represents one of the other sides. Let's call this distance "a". The height up the wall that the ladder reaches will be the remaining side, which we'll call "b".
According to the problem, the ladder is 13 feet long and is placed 6 feet from the base of the building. Using the Pythagorean theorem, we can set up the following equation:
a^2 + b^2 = 13^2
Since we know that a = 6 (the distance from the base of the building to the ladder), we can substitute it into the equation:
6^2 + b^2 = 13^2
Simplifying:
36 + b^2 = 169
Next, we isolate b^2 by subtracting 36 from both sides of the equation:
b^2 = 169 - 36
b^2 = 133
Finally, we solve for b by taking the square root of both sides of the equation:
b = √133
Therefore, the ladder reaches approximately √133 feet up the wall of the building.