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 that the height that it reaches on the side of the bridge. Find the horizontal and vertical distances spanned by this brace.
solve
x^2 + (x+25)^2 = 125^2
To find the horizontal and vertical distances spanned by the brace, we can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.
Let's designate the horizontal distance as x and the vertical distance as y. According to the problem, the diagonal brace has a length of 125 ft.
From the problem description, we can gather the following information:
1. The diagonal brace has a length of 125 ft.
2. The horizontal distance spanned by the brace is 25 ft longer than the vertical distance.
Now we can set up our equations:
1. Pythagorean theorem: x^2 + y^2 = 125^2
2. Horizontal distance equals vertical distance plus 25: x = y + 25
We can solve this system of equations to find the values of x and y.
Substitute the second equation into the first equation:
(y + 25)^2 + y^2 = 125^2
Expand and simplify:
y^2 + 50y + 625 + y^2 = 15625
Combine like terms:
2y^2 + 50y + 625 = 15625
Subtract 15625 from both sides:
2y^2 + 50y - 15000 = 0
Divide the equation by 2 for simplicity:
y^2 + 25y - 7500 = 0
Now we can solve this quadratic equation using factoring, completing the square, or the quadratic formula. I will use factoring:
(y - 50)(y + 75) = 0
This gives us two possible values for y: y = 50 or y = -75. Since we're dealing with distances, we can ignore the negative value.
Therefore, y = 50 ft.
Substitute this value back into the second equation to find x:
x = y + 25
x = 50 + 25
x = 75 ft
So, the horizontal distance spanned by the brace is 75 ft, and the vertical distance spanned by the brace is 50 ft.