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A stone arch in a bridge forms a parabola described by the equation y = a(x - h)2 + k, where y is the height in feet of the arch above the water, x is the horizontal distance from the left end of the arch, a is a constant, and (h, k) is the vertex of the parabola.

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What is the equation that describes the parabola formed by the arch?

Respuesta :

Answer:

For anyone who needs the explanation:

The equation that describes the parabola formed by the arch: y = -0.071(x-13)^2 + 12

The Width of the arch 8 ft above the water: 15

Step-by-step explanation:

  1. The equation of the arch: y = a(x - h)^2 + k
  • By the picture, we see that the vertex is (13,12). The question states that the vertex is (h,k). So H = 13 and K = 12.

    2. Plug values into equation:

  • H = 13. K = 12.
  • Take another point (besides the vertex) from the picture to plug in for X and Y. We can use (26,0)
  • X = 26. Y= 0.
  • Now we have: 0 = a(26 - 13)^2 + 12

   3. Solve Equation to find "a":

  • 0 = a(26-13)^2 +12
  • First, simplify (26-13). Then, subtract 12 from both sides
  • -12 = a(-13)^2
  • Solve (-13)^2. This equals 169.
  • -12 = a(169)
  • Divide 169 on both sides
  • -0.071 = a

   4. Now rewrite the equation y = a(x - h)^2 + k:

  • a = -0.071
  • h = 13
  • k = 12

y = -0.071(x-13)^2 + 12

To find the width of the arch when the height is 8 ft:

  1. Create equation:
  • y = height in feet of arch above water. In this case it will be 8 ft. So y = 8.
  • 8 = -0.071(x-13)^2 + 12

   2. Find "x":

  • x = horizontal distance from left end of the arch
  • ( "x" will not give the width of the arch yet, but will give the x-value on the right point of the arch, to the right of the vertex when the height(y) = 8 )
  • 8 = -0.071(x-13)^2 + 12
  • Subtract 12 from both sides: -4 = -0.071(x-13)^2
  • Divide -0.071 on both sides: (rounded)56 = (x-13)^2
  • Square root property:
  • 56 squared = 7.5(rounded to nearest tenth)
  • (X-13)^2 squared will cancel out the ^2
  • 7.5 = x-13
  • Add 13 to both sides: 20.5 = x

   3. We found the x-value of the point on the right of the arch:

  • x = 20.5 and height(y) = 8 : (20.5,8)

   4. Find the x-value of the point on the left of the arch:

  • Both x-values will be an equal distance from the vertex (13,12)
  • 20.5 - 13 = 7.5
  • So, the right point is 7.5 units to the right of the vertex
  • 7.5 units to the left of the vertex: (13 - 7.5) = 5.5

Now we have (5.5, 8) for the left point of the arch, and (20.5,8) for the right point of the arch. To find the width(x), do 20.5 - 5.5 =

15

Good job!

At 8 feet, the arch is 15 feet wide.

Ver imagen pruittchristina4

The parabola formed by the arch is described by  y ≈ –0.071·(x - 13)² + 12

The reason why the above equation that describes the parabola formed by the arc is correct is given as follows:

The given parameter of the stone arch bridge are;

The equation of the parabola that represents the stone arch is y = a·(x - h)² + k

The height of the arch above water = y in feet

The horizontal distance from the left end of the arch = x

(h, k) = The vertex of the parabola

The vertex of the given parabola = (13, 12)

The coordinates of the left and right end of the arch = (0, 0), and (0, 26)

Required:

To find the equation of the of the parabola that describes the arc

Solution:

The vertex, (h, k) = (13, 12)

Therefore, h = 13, and k = 12

Plugging in the values gives;

y = a·(x - h)² + k

y = a·(x - 13)² + 12

At the point (0, 0), we have;

0 = a·(0 - 13)² + 12 = a·(-13)² + 12

[tex]a = \dfrac{-12}{13^2} \approx 0.071[/tex]

The equation that describes the parabola formed by the arch is therefore;

  • y ≈ -0.071·(x - 13)² + 12

Learn more about the equation of a parabola here:

https://brainly.com/question/5996128

Ver imagen oeerivona

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