Question

A person desires to reach a point that is$\mathbf{3}\mathbf{.}\mathbf{40}\mathbf{}\mathbf{km}$from her present location and in a direction that is$\mathbf{3}\mathbf{.}\mathbf{50}\mathbf{°}$north of east. However, she must travel along streets that are oriented either north–south or east–west. What is the minimum distance she could travel to reach her destination?

Short answer

The minimum distance traveled to reach destination is$4.47\mathrm{km}$

Step-by-step solution

To understand the concept of scalar projection

This problem refers to scalar projection. In Cartesian coordinates, scalar components are scalar projections in the directions of the coordinate axes. Using this concept, the distance can be calculated by finding the components and adding those components.

The components can be written as

$$\begin{gathered} d_x=d\cos \theta \\ {d}_y=d\sin \theta \end{gathered}$$

Therefore the minimum distance D is given by the following formula.

$$\begin{gathered} D=d_x+d_y=d\cos \theta +d\sin \theta \\ \left(\mathrm{i}\right) \end{gathered}$$

To find the minimum distance traveled

Given are,

$$\begin{gathered} d=3.40\mathrm{km} \\ \theta =35.0° \end{gathered}$$

Substituting the above values in equation (i), the minimum distance D can be written as,

$D=\left[3.40\times \cos \left(35.0\right)\right]+\left[3.40\times \sin \left(35.5\right)\right]$

Thus, $D=4.47\mathrm{km}$