91Ó°ÊÓ

A telephone company offers the following plans. Also given are the piecewise functions that model these plans. Use this information to solve Exercises \(95-96\). Plan \(A\) 30 dollars per month buys 120 minutes. \- Additional time costs 0.30 dollars per minute. $$C(t)=\left\\{\begin{array}{ll}30 & \text { if } 0 \leq t \leq 120 \\\30+0.30(t-120) & \text { if } t>120\end{array}\right.$$ Plan \(B\) 40 dollars per month buys 200 minutes. -Additional time costs 0.30 dollars per minute. $$C(t)=\left\\{\begin{array}{ll}40 & \text { if } 0 \leq t \leq 200 \\\40+0.30(t-200) & \text { if } t>200\end{array}\right.$$ Simplify the algebraic expression in the second line of the piecewise function for plan B. Then use point-plotting to graph the function.

Find the area of the donut-shaped region bounded by the graphs of \((x-2)^{2}+(y+3)^{2}=25\) and \((x-2)^{2}+(y+3)^{2}=36\)

A tangent line to a circle is a line that intersects the circle at exactly one point. The tangent line is perpendicular to the radius of the circle at this point of contact. Write an equation in point-slope form for the line tangent to the circle whose equation is \(x^{2}+y^{2}=25\) at the point \((3,-4)\)

Determine whether the graph of \(x^{2}-y^{3}=2\) is symmetric with respect to the \(y\) -axis, the \(x\) -axis, the origin, more than one of these, or none of these. (Section \(1.3,\) Examples 2 and 3 )

Solve by completing the square: $$ x^{2}-2 x-1=0 $$ (Section P.7, Example 9)

The size of a television screen refers to the length of its diagonal. If the length of an HDTV screen is 28 inches and its width is 15.7 inches, what is the size of the screen to the nearest inch? (Section P.8, Example 8)

If a function is defined by an equation, explain how to find its domain.

Is there a relationship between wine consumption and deaths from heart disease? The table gives data from 19 developed countries. $$\begin{array}{|l|c|cccccc|} \hline \text { Country } & \mathbf{A} & \mathbf{B} & \mathbf{C} & \mathbf{D} & \mathbf{E} & \mathbf{F} & \mathbf{G} \\\ \hline \begin{array}{l} \text { Liters of alcohol from } \\ \text { drinking wine, per } \\ \text { person per year }(x) \end{array} & 2.5 & 3.9 & 2.9 & 2.4 & 2.9 & 0.8 & 9.1 \\ \hline \begin{array}{l} \text { Deaths from heart } \\ \text { disease, per } 100,000 \\ \text { people per year }(y) \end{array} & 211 & 167 & 131 & 191 & 220 & 297 & 71 \\ \hline \end{array}$$ $$\begin{array}{|c|c|c|c|c|c|c|ccccc|c|c|} \hline \text { Country } & \mathbf{H} & \mathbf{I} & \mathbf{J} & \mathbf{K} & \mathbf{L} & \mathbf{M} & \mathbf{N} & \mathbf{O} & \mathbf{P} & \mathbf{Q} & \mathbf{R} & \mathbf{S} \\\ \hline(x) & 0.8 & 0.7 & 7.9 & 1.8 & 1.9 & 0.8 & 6.5 & 1.6 & 5.8 & 1.3 & 1.2 & 2.7 \\ \hline \mathbf{( y )} & 211 & 300 & 107 & 167 & 266 & 227 & 86 & 207 & 115 & 285 & 199 & 172 \\ \hline \end{array}$$ a. Use the statistical menu of your graphing utility to enter the 19 ordered pairs of data items shown in the table. b. Use the scatter plot capability to draw a scatter plot of the data. c. Select the linear regression option. Use your utility to obtain values for \(a\) and \(b\) for the equation of the regression line, \(y=a x+b .\) You may also be given a correlation coefficient, \(r .\) Values of \(r\) close to 1 indicate that the points can be described by a linear relationship and the regression line has a positive slope. Values of \(r\) close to \(-1\) indicate that the points can be described by a linear relationship and the regression line has a negative slope. Values of \(r\) close to 0 indicate no linear relationship between the variables. In this case, a linear model does not accurately describe the data. d. Use the appropriate sequence (consult your manual) to graph the regression equation on top of the points in the scatter plot.

You will be developing functions that model given conditions. A company that manufactures bicycles has a fixed cost of \(\$ 100,000 .\) It costs \(\$ 100\) to produce each bicycle. The total cost for the company is the sum of its fixed cost and variable costs. Write the total cost, \(C,\) as a function of the number of bicycles produced, \(x .\) Then find and interpret \(C(90)\).

Describe a procedure for finding \((f \circ g)(x) .\) What is the name of this function?