To find out how far from the base of the pole each bracket should be braced in the ground, we can use the Pythagorean theorem. The pole, the ground distance from the pole to the bracket, and the bracket itself form a right triangle.
Let:
- \( a \) be the height of the pole where the bracket attaches (4.5 ft),
- \( b \) be the ground distance from the base of the pole to the point where the bracket is secured,
- \( c \) be the length of the bracket (6.5 ft).
According to the Pythagorean theorem:
\[ c^2 = a^2 + b^2 \]
Substituting the known values:
\[ (6.5)^2 = (4.5)^2 + b^2 \]
Calculating each term:
\[ 42.25 = 20.25 + b^2 \]
Now, solve for \( b^2 \):
\[ b^2 = 42.25 - 20.25 \] \[ b^2 = 22 \]
Taking the square root of both sides to find \( b \):
\[ b = \sqrt{22} \approx 4.7 \text{ ft} \]
Thus, the distance from the base of the pole that each bracket should be braced in the ground is approximately 4.7 ft.