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:
- <u>The equation of the arch:</u><u> y = a(x - h)^2 + k</u>
- 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. <u>Plug values into equation:</u>
- 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. <u>Solve Equation to find "a":</u>
- 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. <u>Now rewrite the equation y = a(x - h)^2 + k:</u>
<h2>y = -0.071(x-13)^2 + 12</h2><h2 />
To find the width of the arch when the height is 8 ft:
- <u>Create equation:</u>
- 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. <u>Find "x":</u>
- 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
- <u>Square root property:</u>
- 56 squared = 7.5(rounded to nearest tenth)
- (X-13)^2 <em>squared</em> will cancel out the ^2
- 7.5 = x-13
- Add 13 to both sides: 20.5 = x
3. <u>We found the x-value of the point on the </u><em><u>right</u></em><u> of the arch:</u>
- x = 20.5 and height(y) = 8 : (20.5,8)
4. <u>Find the x-value of the point on the </u><em><u>left</u></em><u> of the arch:</u>
- Both x-values will be an equal distance from the vertex (13,12)
- 20.5 - 13 = 7.5
- So, the <em>right</em> point is 7.5 units to the <em>right</em> of the vertex
- 7.5 units to the <em>left</em> of the vertex: (13 - 7.5) = 5.5
Now we have (5.5, 8) for the <em>left</em> point of the arch, and (20.5,8) for the <em>right </em>point of the arch. To find the width(x), do 20.5 - 5.5 =
<h2>
15</h2>
Good job!
At 8 feet, the arch is 15 feet wide.