Chapter 6.2, Problem 58E

(a)

To determine

Towrite:The equation giving the relationship between $d$ and $\theta$ .

Explanation of Solution

Apply the law of cosine.

  $\begin{array}{c}\cos \theta =\frac{7^2+{10}^2-{\left(d\right)}^2}{2\left(7\right)\left(10\right)}\\ \cos \theta =\frac{149-d^2}{140}\\ d=\sqrt{149-140\cos \theta }\end{array}$

Therefore, the required relationship is $d=\sqrt{149-140\cos \theta }$ .

(b)

To determine

To write:The equation for $\theta$ as a function of $d$ .

Explanation of Solution

Apply the law of cosine.

  $\begin{array}{c}\cos \theta =\frac{7^2+{10}^2-{\left(d\right)}^2}{2\left(7\right)\left(10\right)}\\ \cos \theta =\frac{149-d^2}{140}\\ \theta ={\cos }^{-1}\left(\frac{149-d^2}{140}\right)\end{array}$

Therefore, the value of $\theta$ is $\theta ={\cos }^{-1}\left(\frac{149-d^2}{140}\right)$ .

(c)

To determine

Towrite:The equation for $s$ as a function of $\theta$ .

Explanation of Solution

The equation for $x$ is,

  $\begin{array}{c}Lenght=Radius\times Anglesubtended\\ s=4\left(\pi -\theta \right)\end{array}$

Therefore, the expression for $s$ is $s=4\left(\pi -\theta \right)$ .

(d)

To determine

Tocomplete:The table.

Explanation of Solution

The expression for $s$ is,

  $s=4\left(\pi -\theta \right)$ .

The expression for $\theta$ is,

  $\theta ={\cos }^{-1}\left(\frac{149-d^2}{140}\right)$

The complete table is shown below.

d(inches)	$0$	$10$	$12$	$13$	$14$	$15$	$16$
$\theta \left(\text{degree}\right)$	$60.94$	$69.51$	$87.95$	$98.21$	$109.62$	$122.88$	$139.84$
$s\left(\text{inches}\right)$	$478.24$	$441.96$	$368.20$	$327.18$	$281.52$	$228.48$	$160.64$

Table (1)

Therefore, the complete filled table is shown above.