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 far from the base of the pole should each bracket be braced in the ground? Round to the nearest tenth.(1 point) Responses 22.0 ft. 22.0 ft. 4.7 ft. 4.7 ft. 7.9 ft. 7.9 ft. 62.5 ft.

Answer 1

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

The problem gives us:
- The length of the bracket (hypotenuse, \(c\)) = 6.5 ft
- The vertical distance from the ground to the attachment point on the pole (one leg of the right triangle, \(a\)) = 4.5 ft.

We need to find the horizontal distance from the base of the pole to the point where the bracket is braced into the ground (the other leg of the right triangle, \(b\)).

Using the Pythagorean theorem:

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

Plugging in the values:

\[
(6.5)^2 = (4.5)^2 + b^2
\]

Calculating each square:

\[
42.25 = 20.25 + b^2
\]

Now, isolate \(b^2\):

\[
b^2 = 42.25 - 20.25
\]
\[
b^2 = 22
\]

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

\[
b = \sqrt{22} \approx 4.69
\]

Rounding to the nearest tenth, \(b \approx 4.7\) ft.

Thus, the answer is **4.7 ft.**