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Chapter 1: Problem 57
Find a mathematical model that represents the statement. (Determine the constant of proportionality.) \(y\) is inversely proportional to \(x .(y=7 \text { when } x=4 .)\)
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Data Analysis: Light Intensity A light probe is located \(x\) centimeters from a light source, and the intensity \(y\) (in microwatts per square centimeter) of the light is measured. The results are shown as ordered pairs \((x, y)\) (Spreadsheet at LarsonPrecalculus,com) $$\begin{array}{lll} (30,0.1881) & (34,0.1543) & (38,0.1172) \\ (42,0.0998) & (46,0.0775) & (50,0.0645) \end{array}$$ A model for the data is \(y=262.76 / x^{2.12}\) A. Use a graphing utility to plot the data points and the model in the same viewing window. B. Use the model to approximate the light intensity 25 centimeters from the light source.
Match the data with one of the following functions $$f(x)=c x, g(x)=c x^{2}, h(x)=c \sqrt{|x|}, \quad \text {and} \quad r(x)=\frac{c}{x}$$ and determine the value of the constant \(c\) that will make the function fit the data in the table. $$\begin{array}{|c|c|c|c|c|c|}\hline x & -4 & -1 & 0 & 1 & 4 \\\\\hline y & -32 & -2 & 0 & -2 & -32 \\\\\hline \end{array}$$
The height \(y\) (in feet) of a baseball thrown by a child is $$y=-\frac{1}{10} x^{2}+3 x+6$$ where \(x\) is the horizontal distance (in feet) from where the ball was thrown. Will the ball fly over the head of another child 30 feet away trying to catch the ball? (Assume that the child who is trying to catch the ball holds a baseball glove at a height of 5 feet.)
Use a graphing utility to graph the function. Be sure to choose an appropriate viewing window. $$k(x)=1 /(x-3)$$
Find the difference quotient and simplify your Answer: $$f(x)=5 x-x^{2}, \quad \frac{f(5+h)-f(5)}{h}, \quad h \neq 0$$
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