test tomm so plz reply back fast
a concrete bridge over a river has an underside in the shape of a parabolic arch. At water level, the arch is 30 mwide. it has maximum heigt of 10m above the water. the minimum vertical thickness of the concrete is 1.5m.
1-find the algebraic relation that represents the shape of the arch?
2-what is the vertical thickness of the concrete 3m from the centre of the arch?
3-if the water level rises 2m, how wide will the arch be at this new level?
can u plz explain each answer in detail as soon as possible plz
hahahaahah
thats k i figured it out
Reiny -- Point well taken!
Sure, I would be happy to help you with your questions.
1. To find the algebraic relation that represents the shape of the arch, we can use the standard form of a parabolic equation: y = a(x - h)^2 + k, where (h, k) is the vertex of the parabola and "a" is a constant that affects the shape.
In this case, the arch is symmetric with a maximum height of 10m above the water level. The arch is also 30m wide at water level, which means the distance from one side of the arch to the other is 30m. Therefore, the vertex of the parabola is at the midpoint of the arch, which is (0,10).
Using this vertex and the given information, we can find the value of "a". Since the arch has a minimum vertical thickness of 1.5m, we know that at a distance of half the width (15m) from the vertex, the arch goes down by 1.5m from the maximum height (10m). Plugging these values into the equation, we get:
1.5 = a(15 - 0)^2 + 10
Simplifying the equation, we have:
1.5 = 225a + 10
225a = 1.5 - 10
225a = -8.5
a = -8.5/225
a = -0.03778 (rounded to five decimal places)
Therefore, the algebraic relation that represents the shape of the arch is y = -0.03778x^2 + 10.
2. To find the vertical thickness of the concrete 3m from the center of the arch, we need to substitute the given value of x into the equation. Since the vertex is at (0,10) and the distance 3m is to the right of the center, the x-coordinate will be 3. Plugging this value into the equation, we have:
y = -0.03778(3)^2 + 10
y = -0.33798 + 10
y = 9.66202
Therefore, the vertical thickness of the concrete 3m from the center of the arch is approximately 9.66m.
3. If the water level rises by 2m, we need to find the new width of the arch at this level. Since the arch is symmetric, the width will increase by the same amount on each side of the arch. The maximum height of the arch remains the same.
To find the new width, we can calculate the horizontal distance from the vertex to one of the points where the arch intersects with the water level. We know that the vertex is (0,10) and the new water level is 2m higher, so the water level intersects with the arch at y = 12. Plugging this value into the equation, we have:
12 = -0.03778x^2 + 10
-0.03778x^2 = 12 - 10
-0.03778x^2 = 2
x^2 = 2/(-0.03778)
x^2 = -52.9538
However, this does not yield a real value for x, which means the arch will not intersect with the water level if it rises by 2m. In practical terms, this means the width of the arch will remain the same at the new water level.
I hope this explanation is clear and helpful to you. If you have any further questions, feel free to ask!
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