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

Answer 1

To determine how high the ladder will reach against the house, we can use the Pythagorean Theorem, which states:

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

where:
- \( a \) is the height the ladder reaches on the wall,
- \( b \) is the distance from the base of the ladder to the wall,
- \( c \) is the length of the ladder.

In this case, we have:
- \( c = 12 \) ft (the length of the ladder),
- \( b = 6 \) ft (the distance from the base of the ladder to the wall).

We need to find \( a \):

1. Substitute the known values into the Pythagorean theorem:
\[
a^2 + 6^2 = 12^2
\]

2. Calculate \( 6^2 \) and \( 12^2 \):
\[
a^2 + 36 = 144
\]

3. To find \( a^2 \), subtract 36 from both sides:
\[
a^2 = 144 - 36
\]
\[
a^2 = 108
\]

4. Now, take the square root of both sides to find \( a \):
\[
a = \sqrt{108}
\]

5. To simplify \( \sqrt{108} \):
\[
\sqrt{108} = \sqrt{36 \times 3} = \sqrt{36} \times \sqrt{3} = 6\sqrt{3}
\]

6. Now calculate \( 6\sqrt{3} \) approximately:
\[
\sqrt{3} \approx 1.732
\]
\[
6\sqrt{3} \approx 6 \times 1.732 \approx 10.392
\]

Rounding to the nearest tenth, the height the ladder will reach is approximately:

\[
\text{Height} \approx 10.4 \text{ ft}
\]

Therefore, the ladder will reach about **10.4 ft** high on the house.