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Expert-verified Found in: Page 22 ### Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974 # The accompanying sketch represents a maze of one-way streets in a city in the United States. The traffic volume through certain blocks during an hour has been measured. Suppose that the vehicles leaving the area during this hour were exactly the same as those entering it. What can you say about the traffic volume at the four locations indicated by a question mark? Can you figure out exactly how much traffic there was on each block? If not, describe one possible scenario. For each of the four locations, find the highest and the lowest possible traffic volume.

The lowest possible values are 100, 120, 0, 0, and the highest possible values are 270, 250, 150 and 150.

See the step by step solution

## Step 1: Consider the labeled diagram.

The sum of incoming traffic equals the sum of outgoing traffic.

Consider the diagram to create equations. The equations are,

$\begin{array}{c}300+a=400+b\\ b+c+100=250\\ d+300=a+320\\ 150+120=d+c\end{array}$

Simplify the above equations.

$\begin{array}{l}a=250-c\\ b=150-c\\ d=270-c\end{array}$

## Step 2: Compute the highest and lowest possible traffic volumes.

Consider the equation and substitute the values.

$\begin{array}{l}a=250-c\\ b=150-c\\ d=270-c\end{array}$

The lowest possible traffic volume when, $b=0$

$\begin{array}{l}0=150-c⇒c=150\\ a=250-150⇒a=100\\ d=270-150⇒d=120\end{array}$

The highest possible traffic volume when, $c=0$

$\begin{array}{l}a=250-0⇒a=250\\ b=150-0⇒b=150\\ d=270-0⇒d=270\end{array}$

Hence, the lowest possible values are 100, 120, 0, 0, and the highest possible values are 270, 250, 150 and 150. ### Want to see more solutions like these? 