Chapter 8.1, Problem 57AYU

Finding the Speed of a Truck A state trooper is hidden 30 feet from a highway. One second after a truck passes, the angle $\theta$ between the highway and the line of observation from the patrol car to the truck is measured. See the illustration.

(a) If the angle measures ${15}^{\circ }$, how fast is the truck traveling? Express the answer in feet per second and in miles per hour.

(b) If the angle measures ${20}^{\circ }$, how fast is the truck traveling? Express the answer in feet per second and in miles per hour.

(c) If the speed limit is 55 miles per hour and a speeding ticket is issued for speeds of 5 miles per hour or more over the limit, for what angles should the trooper issue a ticket?

To determine

To find:

a. If the angle measures $15°$, how fast is the truck traveling? Express the answer in feet per second and in miles per hour.

Answer to Problem 57AYU

a. $76.3mi/hr$.

Explanation of Solution

Given:

From the figure the distance between trooper and highway is 30 feet which is opposite to the elevation angle.

Formula used:

$\text{tan}\theta =\frac{\text{opposite side}}{\text{adjacent side}}$

Calculation:

a. Angle of elevation measures $15°$.

So we need to find adjacent side so that we can know how far truck passes.

So we have to use $\tan$ ratio to find the adjacent side,

$\text{tan}{15}^{\circ }=\frac{\text{opposite side}}{\text{adjacent side}}$

$\text{tan}{15}^{\circ }=\frac{30}{\text{adjacent side}}$

$\text{Adjacent side }=\frac{30}{\text{tan}{15}^{\circ }}$

$\text{tan}{15}^{\circ }\approx 0.2679$

$\text{Adjacent side }=\frac{30}{0.2679}\approx 111.96\text{ feet}$.

So in one second the truck travelled a distance of $111.96$ feet so the speed of the truck is $111.96$ feet per second,

1 feet per second $\approx 0.6818$ miles per hour.

For $111.11\text{ feet/sec }\approx 76.3$ miles per hour.

To determine

To find:

b. If the angle measures $20°$, how fast is the truck traveling? Express the answer in feet per second and in miles per hour.

Answer to Problem 57AYU

b. $56.2mi/hr$.

Explanation of Solution

Given:

From the figure the distance between trooper and highway is 30 feet which is opposite to the elevation angle.

Formula used:

$\text{tan}\theta =\frac{\text{opposite side}}{\text{adjacent side}}$

Calculation:

b. Angle of measures ${\text{20}}^{\circ }$.

So we need to find adjacent side so that we can know how far truck passes.

So we have to use $\tan$ ratio to find the adjacent side,

$\text{tan}{20}^{\circ }=\frac{\text{opposite side}}{\text{adjacent side}}$

$\text{tan}{20}^{\circ }=\frac{30}{\text{adjacent side}}$

$\text{Adjacent side }=\frac{30}{\text{tan}{20}^{\circ }}$

$\text{tan}{20}^{\circ }\approx 0.364$

$\text{Adjacent side }=\frac{30}{0.364}\approx 82.42\text{ feet}$.

So in one second the truck travelled a distance of $82.42$ feet so the speed of the truck is $82.42$ feet per second,

1 feet per second $\approx 0.6818$ miles per hour.

For $82.42\text{ feet}/\text{sec }\approx 56.2$ miles per hour.

To determine

To find:

c. If the speed limit is 55 miles per hour and a speeding ticket is issued for speeds of 5 miles per hour or more over the limit, for what angles should the trooper issue a ticket?

Answer to Problem 57AYU

c. A ticket is issued if $\theta <18.8°$.

Explanation of Solution

Given:

From the figure the distance between trooper and highway is 30 feet which is opposite to the elevation angle.

Formula used:

$\text{tan}\theta =\frac{\text{opposite side}}{\text{adjacent side}}$

Calculation:

c. The speed limit is 55 miles per hour and a speeding ticket is issued for speeds of 5 miles per hour or more over the limit.

So we can use $\tan$ ratio to find the angle,

$\text{tan}\theta =\frac{30}{87.99}$

$\text{tan}\theta \approx 0.34$

$\theta ={\text{ tan}}^{-1}\left(0.34\right)$

$\theta \approx {18.8}^{\circ }$

A ticket is issued if $\theta <18.8°$.