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.