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)
To find the distance from the base of the pole where each bracket should be braced in the ground, we can use the Pythagorean theorem. Let's call the distance we are trying to find "x".
According to the problem, we have a right triangle with one leg measuring 4.5 ft (from the ground to the point on the pole where the bracket is attached) and another leg measuring x ft (from the base of the pole to the point where the bracket is braced). The hypotenuse of this triangle is the length of the bracket, which is 6.5 ft.
Using the Pythagorean theorem, we can write the following equation:
x^2 + 4.5^2 = 6.5^2
Simplifying:
x^2 + 20.25 = 42.25
Subtracting 20.25 from both sides:
x^2 = 22
Taking the square root of both sides:
x = √22
Rounding to the nearest tenth, x ≈ 4.7
Therefore, each bracket should be braced in the ground approximately 4.7 ft from the base of the pole.
Is it right?
To determine the distance from the base of the pole where each bracket should be braced in the ground, we can use the concept of similar triangles.
Let's denote the distance from the base of the pole to the point where the bracket is braced as x feet.
According to the information provided, the bracket is 6.5 ft long, and it attaches to a point on the pole that is 4.5 ft from the ground. Therefore, the distance from the ground to the point where the bracket is braced is (4.5 + x) ft.
Using the concept of similar triangles, we can set up the following proportion:
(6.5 ft) / x = (4.5 + x) ft / 4.5 ft
Cross multiplying, we get:
6.5 ft * 4.5 ft = x * (4.5 + x) ft
29.25 ft^2 = 4.5x + x^2 ft^2
Rearranging the equation, we have:
x^2 + 4.5x - 29.25 = 0
To solve for x, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For the equation x^2 + 4.5x - 29.25 = 0, the values of a, b, and c are:
a = 1
b = 4.5
c = -29.25
Plugging these values into the quadratic formula, we have:
x = (-4.5 ± √((4.5)^2 - 4(1)(-29.25))) / (2(1))
Simplifying further, we get:
x = (-4.5 ± √(20.25 + 117)) / 2
x = (-4.5 ± √137.25) / 2
Taking only the positive value since distance cannot be negative, we have:
x = (-4.5 + √137.25) / 2
Calculating this value, we find:
x ≈ 1.1 ft
Therefore, each bracket should be braced in the ground approximately 1.1 feet from the base of the pole.
To determine how far from the base of the pole each bracket should be braced in the ground, we can use the Pythagorean theorem.
Let's assume that the distance from the base of the pole to the point where the bracket is braced in the ground is x ft. The height of the pole can be considered as the hypotenuse of a right triangle.
Using the Pythagorean theorem:
Hypotenuse^2 = Base^2 + Height^2
In this case:
(4.5 ft)^2 = x^2 + (6.5 ft)^2
Simplifying:
20.25 ft^2 = x^2 + 42.25 ft^2
Combine like terms:
62.5 ft^2 = x^2
Now, we can solve for x by taking the square root of both sides:
x = √(62.5 ft^2)
x = 7.9 ft (rounded to the nearest tenth)
Therefore, each bracket should be braced in the ground approximately 7.9 ft from the base of the pole to provide extra support in bad weather and high winds.