Chapter 4.6, Problem 85E

To determine

To write the distance $d$ from the camera to a unit in the parade as a function of the angle $x$ , and to graph the function over the interval $-\frac{\pi }{2}<x<\frac{\pi }{2}$ .

Answer to Problem 85E

The graph represents $d$ as a function of $x$ in the interval $-\frac{\pi }{2}<x<\frac{\pi }{2}$ .

  $d=\frac{27}{\cos x}$

Explanation of Solution

Given information:

A television camera is on a reviewing platform 27 meters from the street on which a parade passes from left to right.

  

(Consider $x$ as negative when a unit in the parade approaches from the left)

Calculation:

Let us consider the points to write $d$ as a function of angle $x$

Let $A$ be the point where the camera is situated.

  $B$ is the point on the street from where the parade will be passing left to right.

  $C$ is the point where particular unit of parade will reach.

So here, the points $A,B$ and $C$ form a right angled triangle.

In triangle $ABC$ ,

Distance of the camera from the particular unit in the parade $\left(AC\right)=d$

Distance of the camera from the street on which parade will pass $\left(AB\right)=27\text{m}$

Angle between $AB$ and $AC=x$

Now, using

  $\begin{array}{c}\cos x=\frac{\text{adjacent}}{\text{hypotenuse}}\\ \cos x=\frac{AB}{AC}\\ \cos x=\frac{27}{d}\\ d=\frac{27}{\cos x}\end{array}$

Now let us, plot $d=\frac{27}{\cos x}$ over the interval $-\frac{\pi }{2}<x<\frac{\pi }{2}$

  

The above graph represents $d$ as a function of $x$ in the interval $-\frac{\pi }{2}<x<\frac{\pi }{2}$ .