1. Find the vertex of the parabola: 4y^2+4y-16x=0
2. Find an equation of the parabola, opening down, with vertex (-3,1) and solution point (4,-5).
3. The main cables of a suspension bridge are 20 meters above the road at the towers and 4 meters above the road at the center. The road is 80 meters long. Vertical cables are spaced every 10 meters. The main cables hang in the shape of a parabola. Find the equation of the parabola. Then, determine how high the main cable is 20 meters from the center.
Please help, I've tried solving the problems many times and still can't get it right.
4y^2+4y-16x=0
y^2 + y = 4 x
y^2 + y + 1/4 = 4 (x + 1/16)
(y+1/2)^2 = 4 (x+16)
vertex at (-16 , -1/2)
(x+3)^2 = a(y-1)
(4+3)^2 = a (-5-1)
49 = -6 a
a = -49/6
x^2 + 6 x + 9 = -49/6 (y-1)
x^2 + 6 x + 54/6 = - 49 y/6 + 49/6
x^2 + 6 x - 5/6 = -49 y/6
6 x^2 + 36 - 5 = -49 y
start at the middle up 4 end up 20 (16 above middle
y = kx^2
16 = k(40)^2
16 = k (4)^2(10^2)
1 = 100 k
k = .01
so , adding the 4 at the middle
y = 4 + .01 x^2
at x = 20
y = 4 + .01 (4)(100) = 8
) The main cables of a suspension bridge are 20 meters above the road at the towers and 4 meters above the road at the center. The road is 80 meters long. Vertical cables are spaced every 10 meters. The main cables hang in the shape of a parabola. Find the equation of the parabola. Then, determine how high th
1. To find the vertex of the parabola, first rewrite the equation in standard form, which is in the form of `(y - k)^2 = 4a(x - h)`. The vertex of the parabola is at the point (h, k).
Step 1: Divide the equation by 4: `y^2 + y - 4x = 0`.
Step 2: Move the constant term to the right side: `y^2 + y = 4x`.
Step 3: Complete the square for the y terms: `y^2 + y + 1/4 = 4x + 1/4`.
Step 4: Rewrite the left side as a perfect square: `(y + 1/2)^2 = 4x + 1/4`.
Step 5: Compare the equation with the standard form equation `(y - k)^2 = 4a(x - h)`.
From the equation, we can see that the vertex is at (-1/2, 0).
2. To find an equation of the parabola, opening down, with a vertex and a solution point, you can use the vertex form of a parabola, which is `(y - k) = a(x - h)^2`. Substitute the coordinates of the vertex into the equation to solve for a. Then substitute the coordinates of the solution point to find the complete equation.
Step 1: Start with the vertex form equation: `(y - k) = a(x - h)^2`.
Step 2: Substitute the coordinates of the vertex (-3, 1):
`(y - 1) = a(x + 3)^2`.
Step 3: Substitute the coordinates of the solution point (4, -5):
`(-5 - 1) = a(4 + 3)^2`.
`-6 = a(7)^2`.
Step 4: Solve for a:
`-6 = 49a`.
`a = -6/49`.
Step 5: Substitute a back into the equation:
`(y - 1) = (-6/49)(x + 3)^2`.
Thus, the equation of the parabola, opening down, with a vertex (-3, 1) and a solution point (4, -5), is `(y - 1) = (-6/49)(x + 3)^2`.
3. To find the equation of the parabola for the suspension bridge, we can use the vertex form of a parabola and the given information about the main cables.
Step 1: Identify the vertex and the focus:
The vertex is at the center of the road, so it is (0, 4).
The focus is 20 meters above the road, so it is (0, 24).
Step 2: Use the vertex and focus to find the equation.
Using the general equation of a parabola in vertex form, we have:
`(y - k) = (1/4a)(x - h)^2`, where the vertex is (h, k).
Substitute the vertex and focus coordinates into the equation:
`(y - 4) = (1/4a)(x - 0)^2`.
Step 3: Solve for a:
Using the focus and the directrix of a parabola, we know that the distance from the vertex to the focus is equal to the absolute value of the distance from the vertex to the directrix.
In this case, the vertex is (0, 4), and the focus is (0, 24). The directrix is 20 meters below the vertex, which means the directrix is at (0, -16).
Using the distance formula, we can calculate the distance:
`distance from vertex to focus = distance from vertex to directrix`.
`20 = 16/a`.
`a = 16/20`.
`a = 4/5`.
Step 4: Substitute a back into the equation:
`(y - 4) = (1/(4/5))(x - 0)^2`.
`(y - 4) = (5/4)(x^2)`.
Thus, the equation of the parabola for the suspension bridge is `(y - 4) = (5/4)(x^2)`.
To determine how high the main cable is 20 meters from the center, substitute the x-coordinate into the equation and solve for y.
`(y - 4) = (5/4)((20)^2)`.
`(y - 4) = (5/4)(400)`.
`(y - 4) = 500`.
`y = 504`.
Therefore, the main cable is 504 meters high 20 meters from the center.