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Motorcycle Deaths Suppose that the fatality rate (deaths per 100 million miles traveled) of motorcyclists is given by \(g(x)\), where \(x\) is the percentage of motorcyclists who wear helmets. Next, suppose that the percentage of motorcyclists who wear helmets at time \(t(t\) measured in years) is \(f(t)\), where \(t=0\) corresponds to the year 2000 . a. If \(f(0)=0.64\) and \(g(0.64)=26\), find \((g \circ f)(0)\), and interpret your result. b. If \(f(6)=0.51\) and \(g(0.51)=42\), find \((g \circ f)(6)\), and interpret your result. c. Comment on the results of parts (a) and (b). Source: NHTSA.

Overcrowding of Prisons The 1980 s saw a trend toward oldfashioned punitive deterrence of crime in contrast to the more liberal penal policies and community-based corrections that were popular in the 1960 s and early \(1970 \mathrm{~s}\). As a result, prisons became more crowded, and the gap between the number of people in prison and the prison capacity widened. The number of prisoners (in thousands) in federal and state prisons is approximated by the function $$ N(t)=3.5 t^{2}+26.7 t+436.2 \quad 0 \leq t \leq 10 $$ where \(t\) is measured in years, with \(t=0\) corresponding to 1983\. The number of inmates for which prisons were designed is given by $$ C(t)=24.3 t+365 \quad 0 \leq t \leq 10 $$ where \(C(t)\) is measured in thousands and \(t\) has the same meaning as before. a. Find an expression that shows the gap between the number of prisoners and the number of inmates for which the prisons were designed at any time \(t\) b. Find the gap at the beginning of 1983 and at the beginning of 1986 . Source: U.S. Department of Justice.

Hotel Occupancy Rate The occupancy rate of the all-suite Wonderland Hotel, located near an amusement park, is given by the function $$ r(t)=\frac{10}{81} t^{3}-\frac{10}{3} t^{2}+\frac{200}{9} t+55 \quad 0 \leq t \leq 11 $$ where \(t\) is measured in months and \(t=0\) corresponds to the beginning of January. Management has estimated that the monthly revenue (in thousands of dollars) is approximated by the function $$ R(r)=-\frac{3}{5000} r^{3}+\frac{9}{50} r^{2} \quad 0 \leq r \leq 100 $$ where \(r\) (percent) is the occupancy rate. a. What is the hotel's occupancy rate at the beginning of January? At the beginning of July? b. What is the hotel's monthly revenue at the beginning of January? At the beginning of July?

Social Security Contributions For wages less than the maximum taxable wage base, Social Security contributions by employees are \(7.65 \%\) of the employee's wages. a. Find an equation that expresses the relationship between the wages earned \((x)\) and the Social Security taxes paid ( \(y\) ) by an employee who earns less than the maximum taxable wage base. b. For each additional dollar that an employee earns, by how much is his or her Social Security contribution increased? (Assume that the employee's wages are less than the maximum taxable wage base.) c. What Social Security contributions will an employee who earns \(\$ 75,000\) (which is less than the maximum taxable wage base) be required to make? Source: Social Security Administration.