A bridge is designed with cables that connect two towers that rises above a roadway. Each cable is modeled by the function, H(x)=1/9000x2-7/15x+500, where x and h(x) are measured in feet.
a. what is the height above the road of a cable at it's lowest point?
b. Find the distance between towers.
To solve this problem, we need to understand that the lowest point of the cable will occur at the vertex of the quadratic function represented by the cable's equation.
a. To find the height above the road at the lowest point, we need to determine the y-coordinate of the vertex of the quadratic function. Recall that for a quadratic function in the form of f(x) = ax^2 + bx + c, the x-coordinate of the vertex is given by the formula x = -b / (2a).
In this case, the equation for the cable is H(x) = (1/9000)x^2 - (7/15)x + 500. Comparing this to the standard form ax^2 + bx + c, we can see that a = 1/9000 and b = -7/15. By substituting these values into the formula for the x-coordinate of the vertex, we get x = -(-7/15) / (2 * 1/9000).
Let's simplify this expression:
x = 7/15 / (2/9000)
x = 7/15 * 9000/2
x = 315,000/30
x = 10,500
Now that we have the x-coordinate of the vertex, we can substitute it back into the original function to find the y-coordinate (height above the road):
H(10,500) = (1/9000)(10,500)^2 - (7/15)(10,500) + 500
H(10,500) = 1,225 - 7,000 + 500
H(10,500) = -5,275 feet
Therefore, the height above the road at the lowest point of the cable is -5,275 feet.
b. To find the distance between the towers, we need to determine the x-values at which the height above the road is zero (where the cable meets the road).
Setting H(x) = 0 in the equation for the cable, we get:
(1/9000)x^2 - (7/15)x + 500 = 0
We can solve this quadratic equation using factoring, completing the square, or the quadratic formula. Let's use the quadratic formula:
x = (-(-7/15) ± √((-7/15)^2 - 4(1/9000)(500)))/(2(1/9000))
x = (7/15 ± √((49/225) - (4/9000)(500)))/(2/9000)
Let's simplify this expression further:
x = (7/15 ± √(49/225 - 2000/9000))/(2/9000)
x = (7/15 ± √(49/225 - 2/9))/(2/9000)
x = (7/15 ± √(49 - 50)/(2/9000))
x = (7/15 ± √(-1)/(2/9000))
Since we have a negative value under the square root, we can conclude that the equation does not have real solutions. This means the cable does not meet the road, and hence the distance between the towers is undefined.
In summary:
a. The height above the road at the lowest point is -5,275 feet.
b. The distance between the towers is undefined.