To solve the problem, we can form a right triangle where:
- One leg represents the height at which the ladder touches the building.
- The other leg represents the distance from the bottom of the ladder to the base of the building.
- The hypotenuse represents the length of the ladder.
Given that the 6-foot ladder touches the side of the building at a height of 5 feet, we can use the Pythagorean theorem:
\[ a^2 + b^2 = c^2 \]
Where:
- \( a \) is the height at which the ladder touches the wall (5 feet),
- \( b \) is the distance from the bottom of the ladder to the wall,
- \( c \) is the length of the ladder (6 feet).
Substituting the known values into the equation:
\[ 5^2 + b^2 = 6^2 \]
Calculating the squares:
\[ 25 + b^2 = 36 \]
Subtracting 25 from both sides:
\[ b^2 = 36 - 25 = 11 \]
Taking the square root gives:
\[ b = \sqrt{11} \approx 3.32 \text{ feet} \]
Now, let's examine the 15-foot ladder and determine how high it would touch the wall, denoting the height at which it touches the wall as \( h \).
Using the same distance \( b \) from the wall (since it remains constant for both ladders), we can apply the Pythagorean theorem again:
\[ h^2 + b^2 = 15^2 \]
Substituting \( b^2 \) from earlier, we have:
\[ h^2 + 11 = 225 \]
Subtracting 11 from both sides:
\[ h^2 = 225 - 11 = 214 \]
Taking the square root gives:
\[ h = \sqrt{214} \approx 14.66 \text{ feet} \]
Therefore, the height at which the 15-foot ladder touches the wall is approximately 14.66 feet.
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