Question

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.
Image description
What is the equation that describes the parabola formed by the arch?

Answer 1

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.

Answer 2

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