A 125-ft diagonal brace on a bridge connects a support of the center of the bridge to a side support on the bridge. The horizontal distance that it spans is 25 ft longer than the height that it reaches on the side of the bridge. Find the horizontal and vertical distances spanned by this brace.
125^2=height^2 + horiz^2
125^2=height^2+ (height-25)^2
solve for height first.
To find the horizontal and vertical distances spanned by the brace, we can use the Pythagorean theorem, which relates the lengths of the sides of a right triangle. In this case, the diagonal brace forms a right triangle with the vertical distance on the side of the bridge and the horizontal distance it spans.
Let's use the variables h and x to represent the height and the length of the horizontal distance, respectively. The problem states that the horizontal distance spans 25 ft longer than the height, so we have the equation x = h + 25.
Now, let's apply the Pythagorean theorem:
(diagonal)^2 = (vertical)^2 + (horizontal)^2
Since the diagonal brace has a length of 125 ft, we can rewrite the equation as:
125^2 = h^2 + x^2
Substituting x = h + 25, we get:
125^2 = h^2 + (h + 25)^2
Expanding the equation, we have:
15625 = h^2 + h^2 + 50h + 625
Combining like terms, we have:
2h^2 + 50h + 625 - 15625 = 0
Simplifying the equation, we get:
2h^2 + 50h - 15000 = 0
Now we can use the quadratic formula to find the values of h. The quadratic formula is given by:
h = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 2, b = 50, and c = -15000. Plugging in these values, we get:
h = (-50 ± √(50^2 - 4(2)(-15000))) / (2(2))
Now, we can solve this equation to find the value of h.