Chapter 4.8, Problem 31E

a.

To determine

Find the length l of the zone.

Answer to Problem 31E

  $\boxed{250ft}$

Explanation of Solution

Given information:

A police department has set up a speed enforcement zone on a straight length of highway. A patrol car is parked parallel to the zone, $200$  feet from one end and $150$ feet from the other end.

Find the length l of the zone and the measures of angles $A$ and $B$ (in degrees).

Calculation We shall use the fact that a speed enforcement zone is a straight highway of length $l$ , a patrol car is parked parallel to the zone $200ft$ from one end and $150ft$ from another end.

Let $\angle DFE=\theta$ in the right $\Delta DEF$ . Then the following figure can be drawn.

Now in the right $\Delta DEF$ the length of the zone is given by the Pythagoras theorem as

  $l^2={200}^2+{150}^2$

  $\begin{array}{l}l=\sqrt{{200}^2+{150}^2}\\ =\sqrt{40000+22500}\end{array}$

  $=\sqrt{62500}$

  $l=250ft$

Hence, the length of the zone is given by $\boxed{250ft}$

b.

To determine

Find the minimum amount of time for a vehicle to pass through the zone.

Answer to Problem 31E

  $\boxed{4.87\sec }$

Explanation of Solution

Given information:

A police department has set up a speed enforcement zone on a straight length of highway. A patrol car is parked parallel to the zone, $200$  feet from one end and $150$ feet from the other end.

Find the minimum amount of time (in seconds) it takes for a vehicle to pass through the zone without exceeding the posted speed limit of $35$ miles per hour.

Calculation:

Now,

  $\angle A+\angle B+\angle DEF=180°$ (linear angle)

  $\angle A+\angle B=90°\mathrm{.....}\left(1\right)$

Since, $DF$ is parallel to line $m$ with $FE$ as transverse, the following two alternate angles will be equal.

  $\theta =\angle B$

Now in $\Delta DEF$

  $\cos \theta =\frac{Adjacent}{\text{Hypotenuse}}$

  $=\frac{150}{250}$

  $=0.6$

  $\Rightarrow \theta =Arc\cos \left(0.6\right)$

  $\Rightarrow \theta =53.13$

  $\boxed{\angle B=53.13}$

From (1),

  $\angle A=90°-\angle B$

  $\angle A=90°-53.13°$

  $\boxed{\angle A=36.87°}$

The postal speed of vehicle passing through the zone is $35mph=51.33ft/s$

Distance covered by vehicles= length of zone that is $250ft$ .

  $speed=\frac{dis\tan ce}{time}$

  $time=\frac{dis\tan ce}{speed}$

  $=\frac{250}{53.13}$

  $=4.87\sec$

Hence, minimum amount of time it takes for the vehicles to pass through the zone at speed of $35mph$ is $\boxed{4.87\sec }$