Chapter 6.3, Problem 105E

(a)

To determine

To find: The domain of function of tension $\left(T\right)$ with $\theta$ .

Answer to Problem 105E

  $D\in \left(0,\frac{\pi }{2}\right)$

Explanation of Solution

Given: A loaded barge is being towed by two tugboats, and the magnitude of the resultant is 6000 pounds directed along the axis of the barge. Each tow line makes an angle of $\theta$ degree with the axis of the barge.

The component of tension as shown in figure.

  

The resultant along axis of barge is 6000 pounds

So, $2T\cos \theta =6000g$

  $T=29400\sec \theta$

The function tension in form of $\theta$ is $T=29400\sec \theta$ .

Now, find the domain of function.

  $D\in \left(0,\frac{\pi }{2}\right)$

(b)

To determine

To graph: The function of tension $\left(T\right)$ with $\theta$ .

Answer to Problem 105E

  $\begin{array}{ccccccc}\theta & {10}^{\circ }& {20}^{\circ }& {30}^{\circ }& {40}^{\circ }& {50}^{\circ }& {60}^{\circ }\\ T& 29853.5& 31286.8& 33948.2& 38378.9& 45738.3& 58800\end{array}$

Explanation of Solution

Given: A loaded barge is being towed by two tugboats, and the magnitude of the resultant is 6000 pounds directed along the axis of the barge. Each tow line makes an angle of $\theta$ degree with the axis of the barge. $T=29400\sec \theta$

The function of tension $T=29400\sec \theta$

First draw the graph of above function and then complete the table different value of $\theta$

Now complete the table.

  $\begin{array}{ccccccc}\theta & {10}^{\circ }& {20}^{\circ }& {30}^{\circ }& {40}^{\circ }& {50}^{\circ }& {60}^{\circ }\\ T& 29853.5& 31286.8& 33948.2& 38378.9& 45738.3& 58800\end{array}$

(c)

To determine

To graph: The function of tension $\left(T\right)$ with $\theta$ table.

Answer to Problem 105E

  

Explanation of Solution

Given: A loaded barge is being towed by two tugboats, and the magnitude of the resultant is 6000 pounds directed along the axis of the barge. Each tow line makes an angle of $\theta$ degree with the axis of the barge.

Using graphing utility to graph the function, $T=29400\sec \theta$

Where, y-axis represents tension $\left(T\right)$ and x-axis represents angle $\theta$

  

(d)

To determine

To explain: The tension $\left(T\right)$ increase with increase in angle $\theta$ .

Explanation of Solution

Given: A loaded barge is being towed by two tugboats, and the magnitude of the resultant is 6000 pounds directed along the axis of the barge. Each tow line makes an angle of $\theta$ degree with the axis of the barge.

Using graphing utility to graph the function, $T=29400\sec \theta$

Where, y-axis represents tension $\left(T\right)$ and x-axis represents angle $\theta$

  

As shown in graph, this is increasing function.

So, using graph say tension increase as increase in angle.

This is because secant function is increasing function and tension depends on secant.