Question

A surveyor measures the distance across a straight river by the following method (Fig. P3.7). Starting directly across from a tree on the opposite bank, she walks d=100malong the riverbank to establish a baseline. Then she sights across to the tree. The angle from her baseline to the tree is $\theta =35.0^{\circ }$. How wide is the river?

Short answer

The width of the river is 70.0m.

Step-by-step solution

Define vector

A vector is a quantity that has both a direction and a magnitude, and is used to determine the relative location of two points in space.

Step 2: State given data

d = 100 m

$\theta ={35}^{\circ }$

Step 3: Calculate the width of the river

For polar coordinates $(r,\theta )$, the Cartesian coordinates are $x=r\cos \theta ,y=r\sin \theta$,

If the angle is measured relative to positive x axis $r=\sqrt{X^2+y^2},\theta ={\tan }^1\frac{y}{x}$.

Let the width of the river be y

and the length of the river bed.

Then,

$\begin{array}{l}tan\theta =\frac{y}{d}\\ y=d\tan \theta \\ y=100\times t\mathrm{an}{\left(35.0\right)}^{\circ }\\ =70.0m\end{array}$

Hence the width of the river is 70 m.