Chapter 6.2, Problem 56E

a.

To determine

Use the Law of Cosines to write an equation giving the relationship between $x$ and $\theta$ .

Answer to Problem 56E

  $\boxed{x^2-3x\cos \theta -46.75=0}$

Explanation of Solution

Given information:

An engine has a seven-inch connecting rod fastened to a crank.

  

Use the Law of Cosines to write an equation giving the relationship between $x$ and $\theta$ .

Calculation:

The relationship between $x$ and $\theta$ , now use the Law of Cosines which states that,

  $\begin{array}{l}{a}^2=b^2+c^2-2bc\cos A\\ b^2=a^2+c^2-2ac\cos B\\ c^2=a^2+b^2-2ab\cos C\end{array}$

Now apply this law on the triangle shown in figure,

  $\begin{array}{l}{7}^2={1.5}^2+x^2-2\left(1.5\right)x\cos \theta \\ 49=2.25+x^2-3x\cos \theta \\ x^2-3x\cos \theta -46.75=0\end{array}$

The equation appears to be in quadratic form

  $x^2-3x\cos \theta -46.75=0$

Hence, the equation is, $\boxed{x^2-3x\cos \theta -46.75=0}$

b.

To determine

Write $x$ as a function of $\theta$ .

Answer to Problem 56E

  $\boxed{x=\frac{3\cos \theta +\sqrt{9{\cos }^2\theta +187}}{2}}$

Explanation of Solution

Given information:

An engine has a seven-inch connecting rod fastened to a crank.

  

Write $x$ as a function of $\theta$ . (Select the sign that yields positive values of $x$ .)

Calculation:

Solve the equation for value of $x$ ,

  $x^2-3x\cos \theta -46.75=0$

  $x=\frac{3\cos \theta +\sqrt{9{\cos }^2\theta +187}}{2}$

Taking only positive value of $x$ for $\theta$ .

Hence, $x$ as a function of $\theta$ is, $\boxed{x=\frac{3\cos \theta +\sqrt{9{\cos }^2\theta +187}}{2}}$

c.

To determine

Use a graphing utility to graph the function in part $\left(b\right)$ .

Answer to Problem 56E

  

Explanation of Solution

Given information:

An engine has a seven-inch connecting rod fastened to a crank.

  

Use a graphing utility to graph the function in part $\left(b\right)$ .

Calculation:

To graph the function use $TI83$ graphing calculator,

Step1: Press $Y=$ and enter the function,

  

Step2: press GRAPH button, the display is as shown,

  

Hence,the result shown in graph.

d.

To determine

Use the graph in part $\left(c\right)$ to determine the total distance the piston moves in one cycle.

Answer to Problem 56E

  $\boxed{x=8.5}inches$

Explanation of Solution

Given information:

An engine has a seven-inch connecting rod fastened to a crank.

  

Use the graph in part $\left(c\right)$ to determine the total distance the piston moves in one cycle.

Calculation:

To plot the maximum and minimum of the function use $TI83$ graphing calculator,

Step1: Press $SHIFT+TRACE$ and select $4$ for maximum,

  

Step2: press $TRACE$ button, the maximum value will be shown,

  

Hence,the maximum distance of the piston is, $\boxed{x=8.5}inches$