The equation for a support span is given by h = − 1/40(x − 20)^2, where h metres is the distance below the base of a bridge and x metres is the distance from the left side.
a) What is the turning point of the graph?
b) What is the width of the support span?
Your equation represents a parabola with vertex (20,0), that would be the turning point.
So the vertex of (20,0) would be half-way between the endpoints of the width
so when x = 0, h = -(1/40)(400) = -10
when x = 40 , h = -10
(the -10 suggests 10 m below)
So the width is 40 m.
I don't like the wording of this question.
Ahah, thanks so much for your help!
a) The turning point of the graph represents the highest or lowest point on the curve. To find the turning point, we need to determine the coordinates (x, h) where the slope of the graph changes from positive to negative or vice versa.
Given the equation for the support span: h = -1/40(x - 20)^2
We can rewrite the equation in the standard form: h = -1/40(x^2 - 40x + 400)
The turning point of the graph is given by the formula: x = -b/2a, where a and b are the coefficients of the quadratic equation. In this case, a = -1/40 and b = -40.
x = -(-40)/(2*(-1/40))
x = 40/(2/40)
x = 40 * 40/2
x = 1600/2
x = 800
To find the value of h at the turning point, substitute the value of x into the equation:
h = -1/40(800 - 20)^2
h = -1/40(780)^2
h = -1/40(608400)
h = -15210
Therefore, the turning point of the graph is located at (800, -15210).
b) The width of the support span can be determined by finding the distance between the two x-intercepts. Since the equation represents a downward-opening parabola, the x-intercepts will exist when h = 0.
Setting h = 0 in the equation:
0 = -1/40(x - 20)^2
To solve for x, divide both sides by -1/40:
0 = (x - 20)^2
Take the square root of both sides:
√0 = x - 20
0 = x - 20
Therefore, we have x = 20.
The width of the support span is the distance between the two x-intercepts, which is 2 * distance from the left side to the x-intercept.
Width = 2 * (20 - 0)
Width = 2 * 20
Width = 40 meters
Therefore, the width of the support span is 40 meters.
To find the turning point of the graph, we need to determine the x-coordinate of the vertex of the quadratic equation h = − 1/40(x − 20)^2.
a) The x-coordinate of the vertex of a quadratic equation in the form y = a(x - h)^2 + k is given by x = h. In this case, h = 20 since the equation is h = − 1/40(x − 20)^2. Therefore, the turning point of the graph is at x = 20.
b) To find the width of the support span, we need to determine the x-values where the distance below the base of the bridge (h) is 0. In other words, we need to find the x-intercepts of the graph.
Setting h = 0, we have 0 = − 1/40(x − 20)^2. Since a squared quantity is always non-negative, we can conclude that the only way for the equation to equal zero is if x - 20 equals zero.
Simplifying 0 = − 1/40(x − 20)^2, we get 0 = (x - 20)^2. Taking the square root of both sides, we get ±√0 = x - 20. Since the square root of zero is zero, we are left with 0 = x - 20.
Adding 20 to both sides of the equation, we get x = 20. Therefore, the width of the support span is 20 meters.