Astone 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?

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:

The equation of the arch: y = a(x - h)^2 + kBy 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 +12First, simplify (26-13). Then, subtract 12 from both sides-12 = a(-13)^2Solve (-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 = -0.071h = 13k = 12y = -0.071(x-13)^2 + 12

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

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 + 12Subtract 12 from both sides: -4 = -0.071(x-13)^2Divide -0.071 on both sides: (rounded)56 = (x-13)^2Square root property:56 squared = 7.5(rounded to nearest tenth)(X-13)^2 squared will cancel out the ^27.5 = x-13Add 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 = 7.5So, the right point is 7.5 units to the right of the vertex7.5 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.