To find how far from the base of the pole each bracket should be braced in the ground, we can use the Pythagorean theorem. The bracket acts as the hypotenuse of a right triangle, with the vertical leg being the height from the ground to the attachment point on the pole, and the horizontal leg being the distance from the base of the pole.
Given:
- Length of the bracket (hypotenuse) = 6.5 ft
- Height on the pole (vertical leg) = 4.5 ft
We can calculate the horizontal leg (distance from the base of the pole) using the formula of the Pythagorean theorem:
\[ a^2 + b^2 = c^2 \]
Where:
- \(c\) is the length of the bracket (6.5 ft)
- \(a\) is the height from the ground to the attachment point (4.5 ft)
- \(b\) is the horizontal distance from the base of the pole (what we're trying to find)
Rearranging the formula, we get:
\[ b^2 = c^2 - a^2 \]
Now substituting the values:
\[ b^2 = (6.5)^2 - (4.5)^2 \] \[ b^2 = 42.25 - 20.25 \] \[ b^2 = 22 \] \[ b = \sqrt{22} \approx 4.69 \]
Rounding to the nearest tenth gives:
\[ b \approx 4.7 \text{ ft} \]
Thus, the answer is 4.7 ft.