Circles

To get more than an infinitesimal amount of work out of a Carnot engine, we would have to keep the temperature of its working substance below that of the hot reservoir and above that of the cold reservoir by non-infinitesimal amounts. Consider, then, a Carnot cycle in which the working substance is at temperature \(T_{h w}\) as it absorbs heat from the hot reservoir, and at temperature $T_{hw}$ as it expels heat to the cold reservoir. Under most circumstances the rates of heat transfer will be directly proportional to the temperature differences:

$\frac{Q_h}{\mathrm{\Delta }t}=K\left(T_h-T_{hw}\right)\text{ and }\frac{Q_c}{\mathrm{\Delta }t}=K\left(T_{cw}-T_c\right)$

I've assumed here for simplicity that the constants of proportionality $\left(K\right)$ are the same for both of these processes. Let us also assume that both processes take the

same amount of time, so the $\mathrm{\Delta }{t}^{\mathrm{\prime }}\mathrm{s}$ are the same in both of these equations. 

(a) Assuming that no new entropy is created during the cycle except during the two heat transfer processes, derive an equation that relates the four temperatures $T_h,T_c,T_{hw},\text{ and }{T}_{cw}$

(b) Assuming that the time required for the two adiabatic steps is negligible, write down an expression for the power (work per unit time) output of this engine. Use the first and second laws to write the power entirely in terms of the four temperatures (and the constant $\left(K\right)$), then eliminate $T_{cw}$ using the result of part (a).

(c) When the cost of building an engine is much greater than the cost of fuel (as is often the case), it is desirable to optimize the engine for maximum power output, not maximum efficiency. Show that, for fixed $T_h\text{ and }{T}_c$, the expression you found in part (b) has a maximum value at $T_{hw}=\frac{1}{2}\left(T_h+\sqrt{T_h{T}_c}\right)$. (Hint: You'll have to solve a quadratic equation.) Find the corresponding expression for $T_{CW}$

Name three radii of  $\odot o$.

Draw a circle and several parallel chords. What do you think is true of the midpoints of all such chords? 

Name the diameter of $\odot o$.

Draw a circle with center $O$ and a line $\overleftrightarrow{TS}$ tangent to at $\odot O$. Draw $\overline{OT}$ and use a protractor to find $m\angle OTS$? 

Consider $\overleftrightarrow{RS}$ and $\overleftrightarrow{RS}$. Which is a chord and which is a secant?

 a. Draw a right triangle inscribed in a circle. 

 b. What do you know about the midpoint of the hypotenuse?

 c. Where is the center of the circle?

 d. If the legs of the right triangle are $6$ and $8$. Find the radius of the circle.

Why $\overline{TK}$ is not a chord of $\odot o$.

Plane Z passes through the center of sphere Q.

1. Explain why $QR=QS=QT$.
2. Explain why the intersection of the plane and the sphere is a circle. (The intersection of a sphere with any plane passing through the center of the sphere is called a great circle of the sphere).

Name the tangent of $\odot o$.

What name is given to point of $\odot o$.

Name the line tangent to sphere.

Name a secant of the sphere and a chord of the sphere.

Name the 4 radii (none are drawn in the diagram).

What is the diameter of the circle with radius $8,5.2,4\sqrt{3},j$.

What is the radius of sphere with diameter $14,13,5.6,6n$.

The radii of two concentric circles are $15$cm and $7$cm. A diameter $\overline{AB}$ of the larger circle intersects the smaller circle at $C$ and $D$. Find two possible values for $AC$.

For each exercise draw a circle and inscribe the polygon in the circle.

a. Rectangle.

For each exercise draw a circle and inscribe the polygon in the circle.

   b. A trapezoid. 

For each exercise draw a circle and inscribe the polygon in the circle.

 c. An obtuse triangle.

For each exercise draw a circle and inscribe the polygon in the circle.

  d. A parallelogram. 

For each exercise draw a circle and inscribe the polygon in the circle.

 e. An acute isosceles triangle. 

For each exercise draw a circle and inscribe the polygon in the circle.

 f. A quadrilateral $PQRS$. 

For each exercise draw $\odot O$ with radius $12$. Then draw radii $\overline{OA}$ and to $\overline{OB}$ form an angle with the measure named. Find the length of $\overline{AB}$.

 a. $m\angle AOB=90°$

For each exercise draw $\odot O$ with radius $12$. Then draw radii $\overline{OA}$ and to $\overline{OB}$ form an angle with the measure named. Find the length of $\overline{AB}$.

 b. $m\angle AOB=180°$

For each exercise draw $\odot O$ with radius $12$. Then draw radii $\overline{OA}$ and to $\overline{OB}$ form an angle with the measure named. Find the length of $\overline{AB}$.

 c. $m\angle AOB=60°$

For each exercise draw $\odot O$ with radius $12$. Then draw radii $\overline{OA}$ and to $\overline{OB}$ form an angle with the measure named. Find the length of $\overline{AB}$.

 d. $m\angle AOB=120°$

Draw two points $A$ & $B$ and several circles that pass through $A$ and $B$. Locate the centers of these circles. On the basis of your experiment, complete the following statement:

The centers of all circles passing through $A$ and $B$ lie on ______.

Write an argument to support your statement.

$\odot \mathbf{Q}$ and $\odot \mathbf{R}$ are congruent circles the intersect at $\mathbf{C}$ and $\mathbf{D}$.  $\overline{\mathbf{CD}}$ is called the common chord of the circles. 

1. What kind of quadrilateral is $\mathbf{QDRC}$? Why?
2. $\overline{\mathbf{CD}}$ Must be the perpendicular bisector of $\odot \mathbf{Q}$. Why?
3. If $QC=17$ and $QR=30$, find $CD$.

Draw two congruent circles with radii $\mathbf{6}$ each passing through the centre of other and to find length of their common chord.

$\odot P$ and $\odot Q$ have radii $5$ and $7$ and $PQ=6$. Find the length of the common chord $\overline{AB}$. $APBQ$ is a kite and $\overline{PQ}$ is he perpendicular bisector of $\overline{AB}$. Let be the intersection of $\overline{PQ}$ and $\overline{AB}$. Let $PN=x$ and $AN=y$. Write two equations in term of x and y.

Draw a diagram similar to the one shown, but much larger. Carefully draw the perpendicular bisectors of $\overline{AB}$  and $\overline{BC}$.

1. The perpendicular bisect in a point. Where does that point to be?
2. Write an argument that justifies your answer to part (a).

Find the number of odd and even vertices in each network. Imagine travelling each network to see if it can be traced without backtracking.

Find the number of odd and even vertices in each network. Imagine travelling each network to see if it can be traced without backtracking.

Find the number of odd and even vertices in each network. Imagine travelling each network to see if it can be traced without backtracking.

The number of odd vertices will tell you whether or not a network can be   traced without backtracking. Do you see how? If not, read on.

suppose that a given network can be traced without backtracking.

     a. Consider a vertex that is neither the start nor end of a journey through this network. Is such a vertex odd or even?

     b. Now consider the two vertices at the start and finish of a journey through this network. Can both of these vertices be odd? Even?

     c. Can just one of the start and finish vertices be odd?  

The number of odd vertices will tell you whether or not a network can be   traced without backtracking. Do you see how? If not, read on.

Tell why it is impossible to walk across the seven bridges of Koenigsberg  without   crossing any bridge more than once.    

$\overline{JT}$ is tangent to $\odot$$O$ at $T$. Complete.

If $OT=6$ and $JO=10$, then $JT=?$

How many common external tangents can be drawn to the two circles?

$\overline{JT}$ is tangent to $\odot$$O$ at $T$. Complete.

If $OT=6$ and $JT=10$, then $JO=?$

How many common internal tangents can be drawn to each pair of circles in exercise 1 above ?

$\overline{JT}$ is tangent to $\odot$ $O$ at $T$. Complete.

If $m\angle TOJ=60$ and $OT=6$, then $JO=?$

a. Which pair of circles shown above are externally tangent?

b. Which pair are internally tangent? 

$\overline{JT}$ is tangent to $\odot$$O$ at $T$. Complete.

If $JK=9$ and $KO=8$, then $JT=?$

Given $\overline{PA}$ and $\overline{PB}$ are tangents to $\odot o.$

 Use the diagram at the right to explain how the corollary on page $333$

 follows from Theorem $9-1.$                         

In the diagram, which pairs of angles are complementary? Which pairs of angles are supplementary?

The diagram below shows tangent lines and circles. Find $PD$.

$\overline{RS}$ and $\overline{TU}$ are common internal tangents to the circles. If $RZ=4·7$ and $ZU=7·3$, Find $RS$ and $TU$.

$\left(a\right).$ What do you think is true of common external tangents $\overline{AB}$ and $\overline{CD}$$?$prove it.

$\left(b\right).$ Will your results in part $\left(a\right)$ be true if the circles are congruent$?$

$\overline{TR}$ and $\overline{TS}$ are tangents to $\odot$$O$ from $T;$

                                 $m\angle RTS=36$

$\left(a\right).$ Copy the diagram. Draw $\overline{RS}$ and find $m\angle TSR$ and $m\angle TRS.$

$\left(b\right).$ Draw radii $\overline{OS}$ and $\overline{OR}$ and find $m\angle ORS$ and $m\angle OSR$.

$\left(c\right).$ Find $m\angle ROS$.

$\left(d\right).$ Does your result in part $\left(c\right)$ support one of your conclusions about angles in Classroom Exercise $5?$ Explain.