Chapter 6.6, Problem 42E

Calculating Distance A car travels along a straight road, heading east for 1 h, then traveling for 30 min on another road that leads northeast. If the car has maintained a constant speed of 40 mi/h, how far is it from its starting position?

To determine

To find: The distance between the starting and the ending point of the car.

Answer to Problem 42E

The distance between the starting and the ending point of the car is $55.98\text{mi}$.

Explanation of Solution

Given:

The constant speed of car is $40\text{mi/h}$.

The car travel in east direction for $1\text{h}$ and the car travel in north-east direction for $30\text{min}$.

Formula used: Law of Cosines.

$b^2=a^2+c^2-2ac\cos B$.

Calculation:

The distance covered by the car in $1\text{h}$ towards east direction is $40\text{mi}$ and then distance covered by the car in $30\min$ towards north-east direction is $20\text{mi}$,

Let, the angle between east direction and north-east direction is $45°$.

Draw a $\Delta ABC$ with sides $b=25$, $c=15$.

Figure (1)

The sum of the angles on one side of a straight line is $180°$,

The $\angle ABC=180°-45°=135°$.

Substitute $20$ for $a$, $x$ for $b$, $40$ for $c$ and $135°$ for $B$ in law of cosines for finding $AC$,

$\begin{array}{c}{x}^2={\left(20\right)}^2+{\left(40\right)}^2-2\left(20\right)\left(40\right)\cos 135°\\ =400+1600-1600\left(-0.7071068\right)\\ =2000+1131.3708\\ =3131.3708\end{array}$

Simplify the above expression by taking square root to each side,

$\begin{array}{c}\sqrt{x^2}=\sqrt{3131.3708}\\ x=55.98865\\ x\approx 55.98\text{mi}\end{array}$

Thus, the distance between the starting and the ending point of the car is $55.98\text{mi}$.