Chapter 3, Problem 27RE

27. Parabolic Arch Bridge A horizontal bridge is in the shape of a parabolic arch. Given the information shown in the figure, what is the height h of the arch 2 feet from shore?

To determine

To calculate: The height of the arch 2 feet from the shore.

Answer to Problem 27RE

The height of the arch 2 feet from the shore is $3.6$ feet.

Explanation of Solution

Given:

A horizontal bridge is in shape of a parabolic arch.

Formula used:

The general equation of a parabola is

$y=a{x}^2+bx+c$

Calculation:

Let us consider the given bridge as sitting on the $x\text{-axis}$ and centred at the $y\text{-axis}$.

Then the parabola passes through the points $(-10,0),(10,0)$ and $(0,10)$.

We have the equation of the parabola as $y=a{x}^2+bx+c$.

Substituting the above points in the general equation, we can get the values of $a$, $b$ and $c$.

Thus, we have

At point $(-10,0)$

$0=a{(-10)}^2+b(-10)+c$

$\Rightarrow 100a-10b+c=0$-----(1)

At point $(10,0)$

$0=a{(10)}^2+b(10)+c$

$\Rightarrow 100a+10b+c=0$-----(2)

At point $(0,10)$

$10=a{\left(0\right)}^2+b(0)+c$

$\Rightarrow c=10$-----(3)

Thus, using (3) in (1) and (2), we get

$100a-10b+10=0$-----(1)

$100a+10b+10=0$-----(2)

Adding the new equations (1) and (2), we get

$200a+20=0$

$\Rightarrow a=-0.1$

Substituting the value of $a$ in (1), we get

$100(-0.1)-10b+10=0$

$\Rightarrow -10-10b+10=0$

$\Rightarrow b=0$

Thus, the equation of the parabolic arch is

$y=-0.1x^2+10$

Now, we have to find the height, $y$ at $x=8$.

2 feet from the shore means 2 units lesser than the value of $x$ (from the origin).

Thus, we get

$y=-0.1{(8)}^2+10$

$\Rightarrow y=3.6$

Thus, the height of the arch 2 feet from the shore is $3.6$ feet.