A 75ft diagnol brace on a bridge connects a support of the center of the bridge The horizontal distance that it spans is 15ft longer than the hieght that reaches on the side of the bridge.Find the horizontal and vertical distance spanned by this brace
h^2 + (h + 15)^2 = 75^2
To find the horizontal and vertical distances spanned by the brace, let's denote the vertical distance as 'x'.
According to the given information, the horizontal distance that the brace spans is 15ft longer than the height, which means the horizontal distance is 'x + 15'.
Using the Pythagorean theorem, we know that the square of the diagonal brace (75ft) is equal to the sum of the squares of the horizontal and vertical distances.
So, we can write the equation as:
75^2 = (x + 15)^2 + x^2
Now, let's solve this equation step by step:
5625 = (x + 15)^2 + x^2
Expanding the equation:
5625 = x^2 + 30x + 225 + x^2
Combining like terms:
2x^2 + 30x - 5400 = 0
Dividing the equation by 2 to simplify:
x^2 + 15x - 2700 = 0
Now, we can solve this quadratic equation by factoring or using the quadratic formula. Let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In our equation, a = 1, b = 15, and c = -2700. Substituting these values into the formula:
x = (-15 ± √(15^2 - 4 * 1 * -2700)) / (2 * 1)
Simplifying further:
x = (-15 ± √(225 + 10800)) / 2
x = (-15 ± √11025) / 2
x = (-15 ± 105) / 2
Now, let's consider both the positive and negative values:
When x = (-15 + 105) / 2 = 90 / 2 = 45, this is the vertical distance.
And, the horizontal distance is given as the height plus 15 feet:
Horizontal distance = x + 15 = 45 + 15 = 60 feet.
Therefore, the horizontal distance spanned by the brace is 60 feet, and the vertical distance spanned by the brace is 45 feet.