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.