Answer:
- h(x) = 35(1 -(x/175)²)
- 31.54 m
Step-by-step explanation:
You want the equation of a parabolic arch 35 m high and 350 m in width, and you want the height 55 m from center.
Parabola
We can translate and scale the parent quadratic function to give it the desired shape.
The function y = x² has its vertex at the origin, and opens upward. If we reflect it over the x-axis, we have y = -x², and if we translate it upward 1 unit, it becomes ...
y = 1 -x²
This will have a value of y = 0 at x = ±1, so its span is 2 units. So, we now have the equation of a parabolic arch with a height of 1 unit and a span of 2 units.
Scaling
To make the arch have a height of 35 m, we can multiply the equation by 35:
y = 35(1 -x²)
To give it a width of 350 m, we need to scale the function horizontally by a factor of 350/2 = 175. We do this by replacing x with x/175:
y = 35(1 -(x/175)²) . . . . . parabolic arc 35 high and 350 wide.
Writing this as a function, it becomes ...
h(x) = 35(1 -(x/175)²) . . . . . . equation of the arch
Height
When x = 55, the equation gives the height as ...
h(55) = 35(1 -(55/175)²) = 35(1 -(11/35)²) = 35(1104/1225)
h(55) = 31 19/35 ≈ 31.54 . . . . meters
The height of the arch is about 31.54 meters at a point 55 m from center.
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Additional comment
Vertical scaling by a factor of p is obtained by ...
y = p·f(x)
Horizontal scaling by a factor of q is obtained by ...
y = f(x/q)
When the vertical scale factor is negative, the function graph is reflected over the x-axis.
Translation by (h, k) is obtained by ...
y = f(x -h) +k
Above, we have used reflection, translation, and scaling in both directions.
