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?

Answer 1

To solve for the height that the ladder reaches on the house using the Pythagorean Theorem, we'll denote:

• The length of the ladder (\(c\)) as 12 ft,
• The distance from the base of the house to the bottom of the ladder (\(a\)) as 6 ft,
• The height that the ladder reaches on the house (\(b\)) as the unknown we need to calculate.

According to the Pythagorean Theorem:

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

Substituting the known values:

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

Calculating the squares:

\[ 36 + b^2 = 144 \]

Now, subtract 36 from both sides to isolate \(b^2\):

\[ b^2 = 144 - 36 \]

\[ b^2 = 108 \]

Next, take the square root of both sides to find \(b\):

\[ b = \sqrt{108} \]

We can simplify \(\sqrt{108}\):

\[ \sqrt{108} = \sqrt{36 \cdot 3} = \sqrt{36} \cdot \sqrt{3} = 6\sqrt{3} \]

Now, using a calculator, we find \(6\sqrt{3}\):

\[ 6\sqrt{3} \approx 6 \cdot 1.732 \approx 10.392 \]

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

\[ \boxed{10.4} \text{ ft} \]

Answer 2

An electric pole has a support cable that is attached tot he pole 20 ft from the ground. the cable is anchored on the ground 10 ft from the base of the pole. how long is the cable? round to the nearest tenth.

Answer 3

To find the length of the cable, we can use the Pythagorean Theorem. In this scenario:

• The vertical distance from the ground to the point where the cable is attached to the pole (\(a\)) is 20 ft.
• The horizontal distance from the base of the pole to where the cable is anchored on the ground (\(b\)) is 10 ft.
• The length of the cable (\(c\)) is the hypotenuse we want to find.

According to the Pythagorean Theorem:

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

Substituting the known values:

\[ 20^2 + 10^2 = c^2 \]

Calculating the squares:

\[ 400 + 100 = c^2 \]

Adding them together:

\[ 500 = c^2 \]

Now, taking the square root of both sides to find \(c\):

\[ c = \sqrt{500} \]

We can simplify \(\sqrt{500}\):

\[ \sqrt{500} = \sqrt{100 \cdot 5} = \sqrt{100} \cdot \sqrt{5} = 10\sqrt{5} \]

Now, using a calculator, we find \(10\sqrt{5}\):

\[ 10\sqrt{5} \approx 10 \cdot 2.236 \approx 22.36 \]

Rounding to the nearest tenth, the length of the cable is approximately:

\[ \boxed{22.4} \text{ ft} \]

Answer 4

a streetlight post needs some extra support for potential bad weather and high winds. the city wants to install metal support brackets on the light post. each bracket will be braced into the ground and will then attach to a point on the pole that is 4.5 ft from the ground. each bracket is 6.5 ft long. how far from the base of the pole should each bracket be braced in the ground? round to the nearest tenth.

Answer 5

To find out how far from the base of the pole the bracket should be braced in the ground, we can use the Pythagorean Theorem. Here’s how the problem breaks down:

• The length of the bracket (\(c\)) is 6.5 ft (the hypotenuse).
• The vertical distance from the ground to the point where the bracket attaches to the pole (\(b\)) is 4.5 ft.

We need to find the horizontal distance from the base of the pole to where the bracket is anchored in the ground (\(a\)).

Using the Pythagorean Theorem:

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

Substituting the known values:

\[ a^2 + 4.5^2 = 6.5^2 \]

Calculating the squares:

\[ a^2 + 20.25 = 42.25 \]

Now, we can isolate \(a^2\) by subtracting 20.25 from both sides:

\[ a^2 = 42.25 - 20.25 \]

\[ a^2 = 22 \]

Next, we take the square root of both sides to find \(a\):

\[ a = \sqrt{22} \]

Calculating \(\sqrt{22}\):

\[ \sqrt{22} \approx 4.690 \]

Rounding to the nearest tenth, the distance from the base of the pole to where the bracket should be braced in the ground is approximately:

\[ \boxed{4.7} \text{ ft} \]