Chapter 5, Problem 7CCSR

Solution

To identify : The equation the best represents the given situation.

The correct answer is option B.

Given information :

A bridge is supported by a parabolic arch spans a stream of water $180$ feet wide. There must be a clearance of at least $40$ feet over and $100-$ foot channel in the middle of the stream. The origin is placed at water level directly below the center of the arch.

  

The given options are:

  $y=140\left(x+180\right)\left(x-180\right)$

  $y=-\frac{1}{140}\left(x+90\right)\left(x-90\right)$

  $y=-\frac{1}{140}\left(x+40\right)\left(x-40\right)$

  $y=140\left(x+40\right)\left(x-40\right)$

Explanation :

There must be a clearance of at least $40$ feet over and $100-$ feet channel in the middle of the stream. So, the point of clearance will be at $\left(50\text{ f}t,40\text{ f}t\right)$ .

Substitute the coordinates in each of the given equation to determine the correct one.

Option A

  $\begin{array}{c}y=140\left(x+180\right)\left(x-180\right)\\ 40=140\left(50+180\right)\left(50-180\right)\\ 40\ne 4,186,000\end{array}$

Option B

  $\begin{array}{c}y=-\frac{1}{140}\left(x+90\right)\left(x-90\right)\\ 40=-\frac{1}{140}\left(50+90\right)\left(50-90\right)\\ 40=40\end{array}$

Option C

  $\begin{array}{c}y=-\frac{1}{140}\left(x+40\right)\left(x-40\right)\\ 40=-\frac{1}{140}\left(50+40\right)\left(50-40\right)\\ 40\ne -\frac{45}{7}\end{array}$

Option D

  $\begin{array}{c}y=140\left(x+40\right)\left(x-40\right)\\ 40=140\left(50+40\right)\left(50-40\right)\\ 40\ne 126,000\end{array}$

Hence, option B is correct.