Chapter 6.3, Problem 106E

(a)

To determine

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

Answer to Problem 106E

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

Explanation of Solution

Given: To carry a 100-pound cylindrical weight, two people lift on the ends of short ropes that are ties to an eyelet on the top center of the cylinder. Each rope makes an angle of $\theta$ degrees with the vertical.

The component of tension as shown in figure.

  

The resultant along axis of barge is 100 pounds

So, $2T\cos \theta =100g$

  $T=1600\sec \theta [g=30\text{ ft/s]}$

The function tension in form of $\theta$ is $T=1600\sec \theta$ .

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 106E

  $\begin{array}{ccccccc}\theta & {10}^{\circ }& {20}^{\circ }& {30}^{\circ }& {40}^{\circ }& {50}^{\circ }& {60}^{\circ }\\ T& 1624.7& 1702.7& 1847.5& 2088.7& 2489.2& 3200\end{array}$

Explanation of Solution

Given: To carry a 100-pound cylindrical weight, two people lift on the ends of short ropes that are ties to an eyelet on the top center of the cylinder. Each rope makes an angle of $\theta$ degrees with the vertical.

The function of tension $T=1600\sec \theta$

Now complete the table using graphing utility.

  $\begin{array}{ccccccc}\theta & {10}^{\circ }& {20}^{\circ }& {30}^{\circ }& {40}^{\circ }& {50}^{\circ }& {60}^{\circ }\\ T& 1624.7& 1702.7& 1847.5& 2088.7& 2489.2& 3200\end{array}$

(c)

To determine

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

Answer to Problem 106E

  

Explanation of Solution

Given: To carry a 100-pound cylindrical weight, two people lift on the ends of short ropes that are ties to an eyelet on the top center of the cylinder. Each rope makes an angle of $\theta$ degrees with the vertical.

Using graphing utility to graph the function, $T=1600\sec \theta$

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: To carry a 100-pound cylindrical weight, two people lift on the ends of short ropes that are ties to an eyelet on the top center of the cylinder. Each rope makes an angle of $\theta$ degrees with the vertical.

Using graphing utility to graph the function, $T=1600\sec \theta$

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.