Question

Sylvia is replacing a piece of siding on her house. To make the 12 ft. ladder stable, the bottom of the ladder needs to be 6 ft. from the base of her house. Applying the Pythagorean Theorem and rounding to the nearest tenth, how high will the ladder reach so that Sylvia can replace the siding?(1 point)
Responses

180 ft.
180 ft.
10.4 ft.
10.4 ft.
108 ft.
108 ft.
13.4 ft.

Answer 1

To solve for the height that the ladder reaches, we can use the Pythagorean Theorem, which states that in a right triangle, the square of the length of the hypotenuse (the ladder, in this case) is equal to the sum of the squares of the lengths of the other two sides.

Let's denote:

- \( c \) = the length of the ladder (hypotenuse) = 12 ft
- \( a \) = the distance from the base of the ladder to the house = 6 ft
- \( b \) = the height the ladder reaches (the side opposite to the distance from the house)

According to the Pythagorean theorem:

\[ c^2 = a^2 + b^2 \]

Substituting the values:

\[ 12^2 = 6^2 + b^2 \]

Calculating \( 12^2 \) and \( 6^2 \):

\[ 144 = 36 + b^2 \]

Now, isolate \( b^2 \):

\[ b^2 = 144 - 36 \]
\[ b^2 = 108 \]

Now, take the square root to find \( b \):

\[ b = \sqrt{108} \]

Calculating \( \sqrt{108} \):

\[ b \approx 10.39 \]

Rounding to the nearest tenth gives us:

\[ b \approx 10.4 \]

Therefore, the ladder will reach approximately **10.4 ft** high.

So the correct response is **10.4 ft.**