Question

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 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 1

To find how far from the base of the pole each bracket should be braced in the ground, we can use the Pythagorean theorem.

Let's denote:

• The length of the bracket (hypotenuse) as \( c = 6.5 \) ft,
• The height at which the bracket is attached to the pole (one leg of the right triangle) as \( a = 4.5 \) ft,
• The distance from the base of the pole to where the bracket is braced into the ground (the other leg of the right triangle) as \( b \).

According to the Pythagorean theorem:

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

We can rearrange this to solve for \( b \):

\[ b^2 = c^2 - a^2 \] \[ b = \sqrt{c^2 - a^2} \]

Now plug in the values:

\[ b = \sqrt{(6.5)^2 - (4.5)^2} \]

Calculating \( c^2 \) and \( a^2 \):

\[ (6.5)^2 = 42.25 \] \[ (4.5)^2 = 20.25 \]

Now substitute these values back into the equation:

\[ b = \sqrt{42.25 - 20.25} = \sqrt{22} \]

Now calculate \( \sqrt{22} \):

\[ b \approx 4.6904 \]

Rounding to the nearest tenth, we get:

\[ b \approx 4.7 , \text{ft} \]

Thus, each bracket should be braced approximately 4.7 feet from the base of the pole.