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Chapter 2: Problem 48
Write the complex conjugate of the complex number. Then multiply the number by its complex conjugate. $$\sqrt{-15}$$
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The mean salaries \(S\) (in thousands of dollars) of public school classroom teachers in the United States from 2000 through 2011 are shown in the table. $$\begin{array}{|c|c|}\hline \text { Year } & \text { Salary, \(S\) } \\\\\hline 2000 & 42.2 \\\2001 & 43.7 \\\2002 & 43.8 \\\2003 & 45.0 \\\2004 & 45.6 \\\2005 & 45.9 \\\2006 & 48.2 \\\2007 & 49.3 \\\2008 & 51.3 \\\2009 & 52.9 \\\2010 & 54.4 \\\2011 & 54.2 \\\\\hline\end{array}$$ A model that approximates these data is given by $$S=\frac{42.16-0.236 t}{1-0.026 t}, \quad 0 \leq t \leq 11$$ where \(t\) represents the year, with \(t=0\) corresponding to 2000. (a) Use a graphing utility to create a scatter plot of the data. Then graph the model in the same viewing window. (b) How well does the model fit the data? Explain. (c) Use the model to predict when the salary for classroom teachers will exceed \(\$ 60,000\). (d) Is the model valid for long-term predictions of classroom teacher salaries? Explain.
A rectangular playing field with a perimeter of 100 meters is to have an area of at least 500 square meters. Within what bounds must the length of the rectangle lie?
Use the position equation $$s=-16 t^{2}+v_{0} t+s_{0}$$ where \(s\) represents the height of an object (in feet), \(v_{0}\) represents the initial velocity of the object (in feet per second), \(s_{0}\) represents the initial height of the object (in feet), and \(t\) represents the time (in seconds). A projectile is fired straight upward from ground level \(\left(s_{0}=0\right)\) with an initial velocity of 128 feet per second. (a) At what instant will it be back at ground level? (b) When will the height be less than 128 feet?
The revenue and cost equations for a product are \(R=x(50-0.0002 x)\) and \(C=12 x+150,000,\) where \(R\) and \(C\) are measured in dollars and \(x\) represents the number of units sold. How many units must be sold to obtain a profit of at least \(\$ 1,650,000 ?\) What is the price per unit?
Find all the zeros of the function. When there is an extended list of possible rational zeros, use a graphing utility to graph the function in order to disregard any of the possible rational zeros that are obviously not zeros of the function. $$f(x)=x^{3}+24 x^{2}+214 x+740$$
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