Q.52 - Precalculus Enhanced with Graphing Utilities

Question

Suppose that a fire truck is parked in

front of a building as shown in the figure.

The beacon light on top of the fire truck is located 10 feet from the wall and has a light on each side. If the beacon light rotates 1 revolution every 2 seconds, then a model for determining the distance d, in feet, that the beacon of light is from point A on the wall after t seconds is given by

$d\left(t\right)=|10 \tan (\mathrm{\pi t}\left)\right|$

(a) Graph $d\left(t\right)=|10 \tan (\mathrm{\pi t}\left)\right|$ for $0\le t\le 2$

(b) For what values of t is the function undefined? Explain

what this means in terms of the beam of light on the

wall.

(c) Fill in the following table.

Step-by-Step Solution

Answer

(a) the graph of a function for distance $d\left(t\right)=|10 \tan (\mathrm{\pi t}\left)\right|$ in the interval $0\le t\le 2$ is

(b) for $t=\frac{1}{2},\frac{3}{2}$, the function is undefined which state that the light is facing the wall at a right angle

(c) Table is

(d) The first difference is$32.5,40.2,64.9,170.2$

(e) The first difference state the speed of beacon light which increases with distance.

Step 1. given data

The given function is

$d\left(t\right)=|10 \tan (\mathrm{\pi t}\left)\right|$

Step 2. Part (a)

Plot the graph of the function $d\left(t\right)=|10 \tan (\mathrm{\pi t}\left)\right|$ in the interval $0\le t\le 2$

Step 3. Part (b)

As the tangent function and tangent function is not defined for $\frac{k\mathrm{\pi }}{2}$, where k is an integer

So the function $d\left(t\right)=|10 \tan (\mathrm{\pi t}\left)\right|$ is not defined for

$$\begin{gathered} \mathrm{\pi t}=\frac{\mathrm{k\pi }}{2} \\ \mathrm{t}=\frac{\mathrm{k}}{2} \end{gathered}$$

Where k is an integer

So in interval$0\le t\le 2$, the function is not defined at

 $t=\frac{1}{2},\frac{3}{2}$

So at time$t=\frac{1}{2},\frac{3}{2}$, the light will be facing the wall at a right angle.

Step 4. Part (c)

Substitute the value of t in the function and determine the function value

Step 5. Pard (d)

Determine the first difference

$$\begin{gathered} \frac{d(0.1)-d\left(0\right)}{0.1-0}=\frac{3.25-0}{0.1} \\ =32.5 \\ \frac{d(0.2)-d(0.1)}{0.2-0.1}=\frac{7.27-3.25}{0.1} \\ =40.2 \\ \frac{d(0.3)-d(0.2)}{0.3-0.2}=\frac{13.76-7.27}{0.1} \\ =64.9 \\ \frac{d(0.4)-d(0.3)}{0.4-0.3}=\frac{30.78-13.76}{0.1} \\ =170.2 \end{gathered}$$

Step 6. Part (e)

The first difference state the speed of beacon light

So as the difference increases, the speed  of beacon light increases