Hello, Universe!
I'm not sure how the "thickness" works, so I'll skip part (b)
A concrete bridge over a river has an underside in the shape of a parabolic arch.
At the water level, the arch is 30m wide.
It has a maximum height of 10m above the water.
The minimum vertical thickness of the concrete is 1.5m
(a) Find an equation that represents the shape of the arch.
(b) What is the vertical thickness of the concrete 3m from the centre of the arch?
(c) If the water level rises 2m, how wide will the arch be at this new level? Code:
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15  15
(a) The parabola opens downward and is symmetric to the yaxis.
. . Its equation is of the form: .$\displaystyle y \:=\:ax^2 + c$
We know that $\displaystyle (0,10)$ is on the parabola.
. . We have: .$\displaystyle 10 \:=\:a(0^2) + c\quad\Rightarrow\quad c = 10$
The equation (so far) is: .$\displaystyle y \:=\:ax^2 + 10$
We know that $\displaystyle (15,0)$ is on the parabola.
. . We have: .$\displaystyle 0 \:=\:a(15^2) + 10\quad\Rightarrow\quad a = \frac{2}{45}$
The equation is: .$\displaystyle \boxed{y \:=\:\frac{2}{45}x^2 + 10}$
(c) If the water level rises 2 m, the bridge looks like this: Code:
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When $\displaystyle y = 2$, we have: .$\displaystyle \frac{2}{45}x^2 + 10 \:=\:2\quad\Rightarrow\quad x^2 = 180\quad\Rightarrow\quad x = \pm6\sqrt{5}$
The width of the bridge will be: .$\displaystyle 2 \times 6\sqrt{5}\:\approx\:\boxed{26.8\text{ meters}}$