To determine how high the ladder will reach on the side of Sylvia's house, we can use the Pythagorean theorem. In this scenario, the ladder forms the hypotenuse of a right triangle, while the distance from the base of the house to the bottom of the ladder is one leg, and the height the ladder reaches on the house is the other leg.
Let:
- \( c \) = length of the ladder = 12 ft
- \( a \) = distance from the base of the house to the bottom of the ladder = 6 ft
- \( b \) = height the ladder reaches on the house (this is what we need to find)
According to the Pythagorean theorem: \[ a^2 + b^2 = c^2 \]
Substituting in the known values: \[ 6^2 + b^2 = 12^2 \] \[ 36 + b^2 = 144 \]
Now, subtract 36 from both sides: \[ b^2 = 144 - 36 \] \[ b^2 = 108 \]
Taking the square root of both sides to find \( b \): \[ b = \sqrt{108} \]
Calculating the square root: \[ b \approx 10.3923 \]
Rounding to the nearest tenth, we find: \[ b \approx 10.4 \text{ ft} \]
Thus, the height the ladder will reach on the house is 10.4 ft.